:: Trees: Connected, Acyclic Graphs
:: by Gilbert Lee
::
:: Received February 22, 2005
:: Copyright (c) 2005-2017 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, FINSET_1, ARYTM_3, CARD_1, SUBSET_1, XBOOLE_0, GLIB_000,
RELAT_2, GLIB_001, TREES_1, ZFMISC_1, FUNCT_1, FINSEQ_1, GRAPH_1, ABIAN,
XXREAL_0, RELAT_1, RCOMP_1, FUNCOP_1, ARYTM_1, WAYBEL_0, TARSKI, PBOOLE,
SETFAM_1, ORDINAL1, NAT_1, GLIB_002, XCMPLX_0;
notations TARSKI, XBOOLE_0, CARD_1, ORDINAL1, NUMBERS, SUBSET_1, SETFAM_1,
DOMAIN_1, XCMPLX_0, ABIAN, XXREAL_0, RELAT_1, FUNCT_1, PBOOLE, FUNCT_2,
FINSEQ_1, NAT_1, FUNCOP_1, GLIB_000, GLIB_001;
constructors DOMAIN_1, CARD_FIL, GLIB_001, VALUED_1, XXREAL_2, WELLORD2,
RELSET_1;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FUNCOP_1,
FINSET_1, XREAL_0, INT_1, CARD_1, GLIB_000, ABIAN, GLIB_001, FUNCT_2,
PARTFUN1;
requirements ARITHM, BOOLE, NUMERALS, REAL, SUBSET;
definitions TARSKI;
equalities FUNCOP_1, ORDINAL1, CARD_1;
expansions TARSKI;
theorems CARD_1, CARD_2, FUNCOP_1, FINSEQ_1, FINSEQ_3, FUNCT_1, FUNCT_2,
GLIB_000, GLIB_001, ABIAN, INT_1, JORDAN12, NAT_1, ORDINAL1, PENCIL_1,
TARSKI, XBOOLE_0, XCMPLX_1, ZFMISC_1, XREAL_1, XXREAL_0, RELSET_1;
schemes NAT_1, SUBSET_1, GLIB_000;
begin :: Definitions
definition
let G be _Graph;
attr G is connected means
:Def1:
for u,v being Vertex of G holds ex W being Walk of G st W is_Walk_from u,v;
end;
definition
let G be _Graph;
attr G is acyclic means
:Def2:
not ex W being Walk of G st W is Cycle-like;
end;
definition
let G be _Graph;
attr G is Tree-like means
G is acyclic & G is connected;
end;
registration
cluster trivial -> connected for _Graph;
coherence
proof
let G be _Graph;
assume G is trivial;
then consider x being Vertex of G such that
A1: the_Vertices_of G = {x} by GLIB_000:22;
now
set W = G.walkOf(x);
let u,v be Vertex of G;
u = x & v = x by A1,TARSKI:def 1;
then W is_Walk_from u,v by GLIB_001:13;
hence ex W being Walk of G st W is_Walk_from u,v;
end;
hence thesis;
end;
end;
registration
cluster trivial loopless -> Tree-like for _Graph;
coherence
proof
let G be _Graph;
assume that
A1: G is trivial and
A2: G is loopless;
now
given W being Walk of G such that
A3: W is Cycle-like;
W.edges() <> {} by A3,GLIB_001:136;
then ex e being object st e in W.edges() by XBOOLE_0:def 1;
hence contradiction by A1,A2,GLIB_000:23;
end;
then
A4: G is acyclic;
reconsider G9=G as trivial _Graph by A1;
G9 is connected;
hence thesis by A4;
end;
end;
registration
cluster acyclic -> simple for _Graph;
coherence
proof
let G be _Graph;
assume
A1: not ex W being Walk of G st W is Cycle-like;
now
given e being object such that
A2: e in the_Edges_of G and
A3: (the_Source_of G).e = (the_Target_of G).e;
set v1 = (the_Source_of G).e;
reconsider v1 as Vertex of G by A2,FUNCT_2:5;
set W = G.walkOf(v1,e,v1);
e Joins v1,v1,G by A2,A3,GLIB_000:def 13;
then len W = 3 by GLIB_001:14;
then W is non trivial by GLIB_001:125;
then W is Cycle-like by GLIB_001:def 31;
hence contradiction by A1;
end;
then
A4: G is loopless by GLIB_000:def 18;
now
let e1,e2,v1,v2 be object;
assume that
A5: e1 Joins v1,v2,G and
A6: e2 Joins v1,v2,G;
A7: e2 Joins v2,v1,G by A6,GLIB_000:14;
A8: v1 <> v2 by A4,A5,GLIB_000:18;
now
set W1 = G.walkOf(v1,e1,v2), W = W1.addEdge(e2);
assume
A9: e1 <> e2;
reconsider W1 as Path of G;
A10: W1.last() = v2 by A5,GLIB_001:15;
then
A11: W.last() = v1 by A7,GLIB_001:63;
A12: len W1 = 3 by A5,GLIB_001:14;
A13: now
let n be odd Element of NAT;
assume that
A14: 1 < n and
A15: n <= len W1;
1+1 <= n by A14,NAT_1:13;
then 2*1 < n by XXREAL_0:1;
then 2*1+1 <= n by NAT_1:13;
then
A16: n = 3 by A12,A15,XXREAL_0:1;
W1 = <*v1,e1,v2*> by A5,GLIB_001:def 5;
hence W1.n <> v1 by A8,A16,FINSEQ_1:45;
end;
W1.first() = v1 & W1.last() = v2 by A5,GLIB_001:15;
then
A17: W1 is open by A8,GLIB_001:def 24;
W1.first() = v1 by A5,GLIB_001:15;
then W.first() = v1 by A7,A10,GLIB_001:63;
then
A18: W is closed by A11,GLIB_001:def 24;
A19: e2 Joins W1.last(), v1, G by A5,A7,GLIB_001:15;
then
A20: W is non trivial by GLIB_001:132;
W1.edges() = {e1} by A5,GLIB_001:108;
then not e2 in W1.edges() by A9,TARSKI:def 1;
hence W is Path-like by A19,A17,A13,GLIB_001:150;
then W is Cycle-like by A18,A20,GLIB_001:def 31;
hence contradiction by A1;
end;
hence e1 = e2;
end;
then G is non-multi by GLIB_000:def 20;
hence thesis by A4;
end;
end;
registration
cluster Tree-like -> acyclic connected for _Graph;
coherence;
end;
registration
cluster acyclic connected -> Tree-like for _Graph;
coherence;
end;
registration
let G be _Graph, v be Vertex of G;
cluster -> Tree-like for inducedSubgraph of G,{v},{};
coherence;
end;
definition
let G be _Graph, v be set;
pred G is_DTree_rooted_at v means
G is Tree-like & for x being Vertex
of G holds ex W being DWalk of G st W is_Walk_from v,x;
end;
registration
cluster trivial finite Tree-like for _Graph;
existence
proof
set V = {1}, E = {};
reconsider S = {} as Function of E,V by RELSET_1:12;
set G = createGraph(V,E,S,S);
take G;
thus thesis;
end;
cluster non trivial finite Tree-like for _Graph;
existence
proof
set V = {0,1}, E = {0}, S = 0 .--> 0, T = 0 .--> 1;
A1: dom T = E by FUNCOP_1:13;
A2: now
let x be object;
assume x in E;
then x = 0 by TARSKI:def 1;
then T.x = 1 by FUNCOP_1:72;
hence T.x in V by TARSKI:def 2;
end;
A3: now
let x be object;
assume x in E;
then x = 0 by TARSKI:def 1;
then S.x = 0 by FUNCOP_1:72;
hence S.x in V by TARSKI:def 2;
end;
reconsider T as Function of E,V by A1,A2,FUNCT_2:3;
dom S = E by FUNCOP_1:13;
then reconsider S as Function of E,V by A3,FUNCT_2:3;
set G = createGraph(V,E,S,T);
A4: the_Edges_of G = E by GLIB_000:6;
the_Target_of G = T by GLIB_000:6;
then
A5: (the_Target_of G).0 = 1 by FUNCOP_1:72;
the_Source_of G = S by GLIB_000:6;
then
A6: (the_Source_of G).0 = 0 by FUNCOP_1:72;
now
given W being Walk of G such that
A7: W is Cycle-like;
now
per cases;
suppose
A8: len W = 3;
set e = W.(1+1), v1 = W.1, v2 = W.(1+2);
A9: W.(1+1) Joins W.1, W.(1+2), G by A8,GLIB_001:def 3,JORDAN12:2;
v1 = v2 by A7,A8,GLIB_001:118;
then
A10: (the_Source_of G).e = v1 & (the_Target_of G).e = v1 or (
the_Source_of G).e = v1 & (the_Target_of G).e = v1 by A9,GLIB_000:def 13;
e in the_Edges_of G by A9,GLIB_000:def 13;
then v1 = 0 & v1 = 1 or v1 = 1 & v1 = 0 by A4,A6,A5,A10,TARSKI:def 1;
hence contradiction;
end;
suppose
A11: len W <> 3;
A12: 3 <= len W by A7,GLIB_001:125;
then 3-1 <= len W - 0 by XREAL_1:13;
then 2*1 in dom W by FINSEQ_3:25;
then W.(2*1) in {0} by A4,GLIB_001:8;
then
A13: W.(2*1) = 0 by TARSKI:def 1;
3 < len W by A11,A12,XXREAL_0:1;
then
A14: 3+1 <= len W by NAT_1:13;
then 2*2 in dom W by FINSEQ_3:25;
then W.(2*2) in {0} by A4,GLIB_001:8;
then W.(2*2) = 0 by TARSKI:def 1;
hence contradiction by A7,A14,A13,GLIB_001:138;
end;
end;
hence contradiction;
end;
then
A15: G is acyclic;
set W1 = G.walkOf(0,0,1), W2 = W1.reverse();
take G;
A16: the_Vertices_of G = V by GLIB_000:6;
then reconsider a0 = 0, a1 = 1 as Vertex of G by TARSKI:def 2;
now
assume card (the_Vertices_of G) = 1;
then ex x being object st the_Vertices_of G = {x} by CARD_2:42;
hence contradiction by A16,ZFMISC_1:5;
end;
hence G is non trivial & G is finite by GLIB_000:def 19;
0 in the_Edges_of G by A4,TARSKI:def 1;
then 0 Joins 0,1, G by A6,A5,GLIB_000:def 13;
then
A17: W1 is_Walk_from 0,1 by GLIB_001:15;
then
A18: W2 is_Walk_from 1,0 by GLIB_001:23;
now
let u,v be Vertex of G;
now
per cases by A16,TARSKI:def 2;
suppose
u = 0 & v = 0;
then G.walkOf(a0) is_Walk_from u,v by GLIB_001:13;
hence ex W being Walk of G st W is_Walk_from u,v;
end;
suppose
u = 0 & v = 1;
hence ex W being Walk of G st W is_Walk_from u,v by A17;
end;
suppose
u = 1 & v = 0;
hence ex W being Walk of G st W is_Walk_from u,v by A18;
end;
suppose
u = 1 & v = 1;
then G.walkOf(a1) is_Walk_from u,v by GLIB_001:13;
hence ex W being Walk of G st W is_Walk_from u,v;
end;
end;
hence ex W being Walk of G st W is_Walk_from u,v;
end;
then G is connected;
hence thesis by A15;
end;
end;
registration
let G be _Graph;
cluster trivial finite Tree-like for Subgraph of G;
existence
proof
set IT = the finite trivial simple Subgraph of G;
take IT;
thus thesis;
end;
end;
registration
let G be acyclic _Graph;
cluster -> acyclic for Subgraph of G;
coherence
proof
let G2 be Subgraph of G;
now
given W2 being Walk of G2 such that
A1: W2 is Cycle-like;
reconsider W = W2 as Walk of G by GLIB_001:167;
A2: W is Path-like by A1,GLIB_001:176;
W is closed & W is non trivial by A1,GLIB_001:176;
then W is Cycle-like by A2,GLIB_001:def 31;
hence contradiction by Def2;
end;
hence thesis;
end;
end;
definition
let G be _Graph, v be Vertex of G;
func G.reachableFrom(v) -> non empty Subset of the_Vertices_of G means
:Def5:
for x being object
holds x in it iff ex W being Walk of G st W is_Walk_from v,x;
existence
proof
defpred P[set] means ex W being Walk of G st W is_Walk_from v,$1;
consider IT being Subset of the_Vertices_of G such that
A1: for x being set holds x in IT iff x in the_Vertices_of G & P[x]
from SUBSET_1:sch 1;
G.walkOf(v) is_Walk_from v,v by GLIB_001:13;
then reconsider IT as non empty Subset of the_Vertices_of G by A1;
take IT;
let x being object;
thus x in IT implies ex W being Walk of G st W is_Walk_from v,x by A1;
assume
A2: ex W being Walk of G st W is_Walk_from v, x;
then x is Vertex of G by GLIB_001:18;
hence thesis by A1,A2;
end;
uniqueness
proof
let IT1, IT2 be non empty Subset of the_Vertices_of G such that
A3: for x being object holds x in IT1 iff ex W being Walk of G st W
is_Walk_from v,x and
A4: for x being object holds x in IT2 iff ex W being Walk of G st W
is_Walk_from v,x;
now
let x be object;
hereby
assume x in IT1;
then ex W being Walk of G st W is_Walk_from v, x by A3;
hence x in IT2 by A4;
end;
assume x in IT2;
then ex W being Walk of G st W is_Walk_from v, x by A4;
hence x in IT1 by A3;
end;
hence thesis by TARSKI:2;
end;
end;
definition
let G be _Graph, v be Vertex of G;
func G.reachableDFrom(v) -> non empty Subset of the_Vertices_of G means
:Def6: for x being object
holds x in it iff ex W being DWalk of G st W is_Walk_from v,x;
existence
proof
set W = G.walkOf(v);
defpred P[set] means ex W being directed Walk of G st W is_Walk_from v,$1;
consider IT being Subset of the_Vertices_of G such that
A1: for x being set holds x in IT iff x in the_Vertices_of G & P[x]
from SUBSET_1:sch 1;
W is_Walk_from v,v by GLIB_001:13;
then reconsider IT as non empty Subset of the_Vertices_of G by A1;
take IT;
let x be object;
thus x in IT implies ex W being directed Walk of G st W is_Walk_from v,x
by A1;
given W being directed Walk of G such that
A2: W is_Walk_from v, x;
x is Vertex of G by A2,GLIB_001:18;
hence thesis by A1,A2;
end;
uniqueness
proof
let IT1, IT2 be non empty Subset of the_Vertices_of G such that
A3: for x being object holds x in IT1 iff ex W being directed Walk of G
st W is_Walk_from v,x and
A4: for x being object holds x in IT2 iff ex W being directed Walk of G
st W is_Walk_from v,x;
now
let x be object;
hereby
assume x in IT1;
then ex W being directed Walk of G st W is_Walk_from v, x by A3;
hence x in IT2 by A4;
end;
assume x in IT2;
then ex W being directed Walk of G st W is_Walk_from v, x by A4;
hence x in IT1 by A3;
end;
hence thesis by TARSKI:2;
end;
end;
Lm1: for G being _Graph, v being Vertex of G holds v in G.reachableFrom(v)
proof
let G be _Graph, v be Vertex of G;
G.walkOf(v) is_Walk_from v,v by GLIB_001:13;
hence thesis by Def5;
end;
Lm2: for G being _Graph, v1 being Vertex of G, e,x,y being object
holds x in G
.reachableFrom(v1) & e Joins x,y,G implies y in G.reachableFrom(v1)
proof
let G be _Graph, v1 be Vertex of G, e,x,y be object;
set RFV = G.reachableFrom(v1);
assume that
A1: x in RFV and
A2: e Joins x,y,G;
consider W being Walk of G such that
A3: W is_Walk_from v1,x by A1,Def5;
W.addEdge(e) is_Walk_from v1,y by A2,A3,GLIB_001:66;
hence thesis by Def5;
end;
Lm3: for G being _Graph, v1,v2 being Vertex of G holds v1 in G.reachableFrom(
v2) implies G.reachableFrom(v1) = G.reachableFrom(v2)
proof
let G be _Graph, v1,v2 be Vertex of G;
assume v1 in G.reachableFrom(v2);
then consider WA being Walk of G such that
A1: WA is_Walk_from v2,v1 by Def5;
A2: WA.reverse() is_Walk_from v1,v2 by A1,GLIB_001:23;
now
let x be object;
hereby
assume x in G.reachableFrom(v1);
then consider WB being Walk of G such that
A3: WB is_Walk_from v1,x by Def5;
WA.append(WB) is_Walk_from v2,x by A1,A3,GLIB_001:31;
hence x in G.reachableFrom(v2) by Def5;
end;
assume x in G.reachableFrom(v2);
then consider WB being Walk of G such that
A4: WB is_Walk_from v2,x by Def5;
WA.reverse().append(WB) is_Walk_from v1,x by A2,A4,GLIB_001:31;
hence x in G.reachableFrom(v1) by Def5;
end;
hence thesis by TARSKI:2;
end;
Lm4: for G being _Graph, W being Walk of G, v being Vertex of G holds v in W
.vertices() implies W.vertices() c= G.reachableFrom(v)
proof
let G be _Graph, W be Walk of G, v be Vertex of G;
assume v in W.vertices();
then consider m being odd Element of NAT such that
A1: m <= len W and
A2: W.m = v by GLIB_001:87;
let x be object;
assume x in W.vertices();
then consider n being odd Element of NAT such that
A3: n <= len W and
A4: W.n = x by GLIB_001:87;
now
per cases;
suppose
m <= n;
then W.cut(m,n) is_Walk_from v,x by A2,A3,A4,GLIB_001:37;
hence ex W2 being Walk of G st W2 is_Walk_from v,x;
end;
suppose
m > n;
then W.cut(n,m) is_Walk_from x,v by A1,A2,A4,GLIB_001:37;
then W.cut(n,m).reverse() is_Walk_from v,x by GLIB_001:23;
hence ex W2 being Walk of G st W2 is_Walk_from v,x;
end;
end;
hence thesis by Def5;
end;
definition
let G1 be _Graph, G2 be Subgraph of G1;
attr G2 is Component-like means
:Def7:
G2 is connected & not ex G3 being connected Subgraph of G1 st G2 c< G3;
end;
registration
let G be _Graph;
cluster Component-like -> connected for Subgraph of G;
coherence;
end;
registration
let G be _Graph, v be Vertex of G;
cluster -> Component-like for inducedSubgraph of G,G.reachableFrom(v);
coherence
proof
let G2 be inducedSubgraph of G,G.reachableFrom(v);
A1: the_Vertices_of G2 = G.reachableFrom(v) by GLIB_000:def 37;
A2: the_Edges_of G2 = G.edgesBetween(G.reachableFrom(v)) by GLIB_000:def 37;
A3: now
A4: v in the_Vertices_of G2 by A1,Lm1;
given G3 being connected Subgraph of G such that
A5: G2 c< G3;
G2 c= G3 by A5,GLIB_000:def 36;
then
A6: G2 is Subgraph of G3 by GLIB_000:def 35;
then
A7: the_Vertices_of G2 c= the_Vertices_of G3 by GLIB_000:def 32;
A8: now
given x being object such that
A9: x in the_Vertices_of G3 and
A10: not x in the_Vertices_of G2;
consider W being Walk of G3 such that
A11: W is_Walk_from v,x by A4,A7,A9,Def1;
reconsider W as Walk of G by GLIB_001:167;
W is_Walk_from v,x by A11,GLIB_001:19;
hence contradiction by A1,A10,Def5;
end;
A12: the_Vertices_of G2 c= the_Vertices_of G3 by A6,GLIB_000:def 32;
now
per cases by A5,GLIB_000:99;
suppose
ex x being object st x in the_Vertices_of G3 & not x in
the_Vertices_of G2;
hence contradiction by A8;
end;
suppose
ex e being object
st e in the_Edges_of G3 & not e in the_Edges_of G2;
then consider e being set such that
A13: e in the_Edges_of G3 and
A14: not e in the_Edges_of G2;
set v1 = (the_Source_of G3).e, v2 = (the_Target_of G3).e;
A15: e Joins v1,v2,G3 by A13,GLIB_000:def 13;
then
A16: e Joins v1,v2,G by GLIB_000:72;
now
per cases;
suppose
the_Vertices_of G3 = the_Vertices_of G2;
then reconsider v1,v2 as Vertex of G2 by A15,GLIB_000:13;
v1 in G.reachableFrom(v) & v2 in G.reachableFrom(v) by A1;
hence contradiction by A2,A14,A16,GLIB_000:32;
end;
suppose
the_Vertices_of G3 <> the_Vertices_of G2;
then the_Vertices_of G2 c< the_Vertices_of G3 by A12,
XBOOLE_0:def 8;
hence contradiction by A8,XBOOLE_0:6;
end;
end;
hence contradiction;
end;
end;
hence contradiction;
end;
now
let x,y be Vertex of G2;
consider W1R being Walk of G such that
A17: W1R is_Walk_from v,x by A1,Def5;
consider W2 being Walk of G such that
A18: W2 is_Walk_from v,y by A1,Def5;
set W1 = W1R.reverse(), W = W1.append(W2);
A19: W1 is_Walk_from x,v by A17,GLIB_001:23;
then
A20: W is_Walk_from x,y by A18,GLIB_001:31;
A21: W1.last() = v by A19,GLIB_001:def 23;
then
A22: v in W1.vertices() by GLIB_001:88;
A23: W.edges() c= G.edgesBetween(W.vertices()) by GLIB_001:109;
W2.first() = v by A18,GLIB_001:def 23;
then W.vertices() = W1.vertices() \/ W2.vertices() by A21,GLIB_001:93;
then
A24: v in W.vertices() by A22,XBOOLE_0:def 3;
then G.edgesBetween(W.vertices()) c= G.edgesBetween(the_Vertices_of G2)
by A1,Lm4,GLIB_000:36;
then W.edges() c= G.edgesBetween(the_Vertices_of G2) by A23;
then reconsider W as Walk of G2 by A1,A2,A24,Lm4,GLIB_001:170;
take W;
thus W is_Walk_from x,y by A20,GLIB_001:19;
end;
then G2 is connected;
hence thesis by A3;
end;
end;
registration
let G be _Graph;
cluster Component-like for Subgraph of G;
existence
proof
set v = the Vertex of G;
set IT = the inducedSubgraph of G,G.reachableFrom(v);
take IT;
thus thesis;
end;
end;
definition
let G be _Graph;
mode Component of G is Component-like Subgraph of G;
end;
definition
let G be _Graph;
func G.componentSet() -> non empty Subset-Family of the_Vertices_of G means
:Def8:
for x being set holds x in it iff ex v being Vertex of G st x = G
.reachableFrom(v);
existence
proof
set v = the Vertex of G;
defpred P[set] means ex v being Vertex of G st $1 = G.reachableFrom(v);
consider IT being Subset-Family of the_Vertices_of G such that
A1: for x being set holds x in IT iff x in bool the_Vertices_of G & P[
x] from SUBSET_1:sch 1;
set x = G.reachableFrom(v);
x in IT by A1;
then reconsider IT as non empty Subset-Family of the_Vertices_of G;
take IT;
thus thesis by A1;
end;
uniqueness
proof
defpred P[object] means ex v being Vertex of G st $1 = G.reachableFrom(v);
let IT1,IT2 be non empty Subset-Family of the_Vertices_of G such that
A2: for x being set holds x in IT1 iff P[x] and
A3: for x being set holds x in IT2 iff P[x];
now
let x be object;
x in IT1 iff P[x] by A2;
hence x in IT1 iff x in IT2 by A3;
end;
hence thesis by TARSKI:2;
end;
end;
registration
let G be _Graph, X be Element of G.componentSet();
cluster -> Component-like for inducedSubgraph of G,X;
coherence
proof
let G2 be inducedSubgraph of G,X;
ex v being Vertex of G st X = G.reachableFrom(v) by Def8;
hence thesis;
end;
end;
definition
let G be _Graph;
func G.numComponents() -> Cardinal equals
card G.componentSet();
coherence;
end;
definition
let G be finite _Graph;
redefine func G.numComponents() -> non zero Element of NAT;
coherence
proof
G.numComponents() = card G.componentSet();
hence thesis;
end;
end;
definition
let G be _Graph, v be Vertex of G;
attr v is cut-vertex means
for G2 being removeVertex of G,v holds G
.numComponents() in G2.numComponents();
end;
definition
let G be finite _Graph, v be Vertex of G;
redefine attr v is cut-vertex means
for G2 being removeVertex of G,v
holds G.numComponents() < G2.numComponents();
compatibility
proof
hereby
assume
A1: v is cut-vertex;
let G2 be removeVertex of G,v;
card Segm G.numComponents() in card Segm G2.numComponents() by A1;
hence G.numComponents() < G2.numComponents() by NAT_1:41;
end;
assume
A2: for G2 being removeVertex of G,v holds G.numComponents() < G2
.numComponents();
now
let G2 be removeVertex of G,v;
G.numComponents() < G2.numComponents() by A2;
then card Segm G.numComponents() in card Segm G2.numComponents()
by NAT_1:41;
hence G.numComponents() in G2.numComponents();
end;
hence thesis;
end;
end;
Lm5: for G1 being non trivial connected _Graph, v being Vertex of G1, G2 being
removeVertex of G1,v st v is endvertex holds G2 is connected
proof
let G1 be non trivial connected _Graph, v being Vertex of G1, G2 be
removeVertex of G1,v;
set VG = the_Vertices_of G1, VG2 = the_Vertices_of G2;
assume
A1: v is endvertex;
then consider ev being object such that
A2: v.edgesInOut() = {ev} and
not ev Joins v, v,G1 by GLIB_000:def 51;
now
let v19,v29 be Vertex of G2;
reconsider v1=v19, v2=v29 as Vertex of G1 by GLIB_000:42;
consider W being Walk of G1 such that
A3: W is_Walk_from v1,v2 by Def1;
set T = the Trail of W;
A4: T is_Walk_from v1, v2 by A3,GLIB_001:160;
then
A5: T.(len T) = v2 by GLIB_001:17;
v19 in VG2;
then v19 in VG \ {v} by GLIB_000:47;
then
A6: not v1 in {v} by XBOOLE_0:def 5;
v29 in VG2;
then v29 in VG \ {v} by GLIB_000:47;
then not v2 in {v} by XBOOLE_0:def 5;
then
A7: v2 <> v by TARSKI:def 1;
A8: T.1 = v1 by A4,GLIB_001:17;
now
let e be object;
assume
A9: e in T.edges();
then consider n being even Element of NAT such that
A10: 1 <= n and
A11: n <= len T and
A12: T.n = e by GLIB_001:99;
n in dom T by A10,A11,FINSEQ_3:25;
then consider n1 being odd Element of NAT such that
A13: n1 = n-1 and
A14: n-1 in dom T and
A15: n+1 in dom T and
A16: T.n Joins T.(n1), T.(n+1),G1 by GLIB_001:9;
A17: n+1 <= len T by A15,FINSEQ_3:25;
A18: n1 <= len T by A13,A14,FINSEQ_3:25;
now
assume
A19: e in v.edgesInOut();
then
A20: e = ev by A2,TARSKI:def 1;
now
per cases by A12,A16,A19,GLIB_000:65;
suppose
A21: T.(n1) = v;
reconsider n2 = n1 - 1 as even Element of NAT by ABIAN:12,INT_1:5;
A22: 1 <= n1 by ABIAN:12;
n1 <> 1 by A6,A8,A21,TARSKI:def 1;
then
A23: 1 < n1 by A22,XXREAL_0:1;
then 1+1 <= n1 by NAT_1:13;
then
A24: 1+1-1 <= n2 by XREAL_1:13;
T.vertexAt(n1) = v by A18,A21,GLIB_001:def 8;
then T.n2 in v.edgesInOut() by A18,A23,GLIB_001:11;
then
A25: T.n = T.n2 by A2,A12,A20,TARSKI:def 1;
n - 1 < n - 0 by XREAL_1:15;
then n1 - 1 < n - 0 by A13,XREAL_1:14;
hence contradiction by A11,A25,A24,GLIB_001:138;
end;
suppose
A26: T.(n+1) = v;
then
A27: n+1 < len T by A7,A5,A17,XXREAL_0:1;
T.vertexAt(n+1) = v by A17,A26,GLIB_001:def 8;
then T.(n+1+1) in v.edgesInOut() by A27,GLIB_001:10;
then
A28: T.n = T.(n+1+1) by A2,A12,A20,TARSKI:def 1;
n + 0 < n+(1+1) & n+1+1 <= len T by A27,NAT_1:13,XREAL_1:8;
hence contradiction by A10,A28,GLIB_001:138;
end;
end;
hence contradiction;
end;
then e in (the_Edges_of G1) \ v.edgesInOut() by A9,XBOOLE_0:def 5;
then e in (the_Edges_of G1) \ G1.edgesInOut({v}) by GLIB_000:def 40;
then e in G1.edgesBetween( the_Vertices_of G1 \ {v}) by GLIB_000:35;
hence e in the_Edges_of G2 by GLIB_000:47;
end;
then
A29: T.edges() c= the_Edges_of G2;
A30: v1 <> v by A6,TARSKI:def 1;
now
let x be object;
assume
A31: x in T.vertices();
now
assume x = v;
then v = T.first() or v = T.last() by A1,A31,GLIB_001:143;
hence contradiction by A30,A7,A4,GLIB_001:def 23;
end;
then not x in {v} by TARSKI:def 1;
then x in VG \ {v} by A31,XBOOLE_0:def 5;
hence x in VG2 by GLIB_000:47;
end;
then T.vertices() c= VG2;
then reconsider W9=T as Walk of G2 by A29,GLIB_001:170;
W9 is_Walk_from v19,v29 by A4,GLIB_001:19;
hence ex W being Walk of G2 st W is_Walk_from v19,v29;
end;
hence thesis;
end;
Lm6: for G being _Graph holds (ex v1 being Vertex of G st for v2 being Vertex
of G ex W being Walk of G st W is_Walk_from v1,v2) implies G is connected
proof
let G be _Graph;
given v1 being Vertex of G such that
A1: for v2 being Vertex of G ex W being Walk of G st W is_Walk_from v1, v2;
now
let x,y be Vertex of G;
consider W1 being Walk of G such that
A2: W1 is_Walk_from v1,x by A1;
consider W2 being Walk of G such that
A3: W2 is_Walk_from v1,y by A1;
W1.reverse() is_Walk_from x,v1 by A2,GLIB_001:23;
then W1.reverse().append(W2) is_Walk_from x,y by A3,GLIB_001:31;
hence ex W being Walk of G st W is_Walk_from x,y;
end;
hence thesis;
end;
Lm7: for G being _Graph holds (ex v being Vertex of G st G.reachableFrom(v) =
the_Vertices_of G) implies G is connected
proof
let G be _Graph;
assume ex v being Vertex of G st G.reachableFrom(v)=the_Vertices_of G;
then consider v being Vertex of G such that
A1: G.reachableFrom(v) = the_Vertices_of G;
now
let x,y be Vertex of G;
consider W1 being Walk of G such that
A2: W1 is_Walk_from v,x by A1,Def5;
consider W2 being Walk of G such that
A3: W2 is_Walk_from v,y by A1,Def5;
W1.reverse() is_Walk_from x,v by A2,GLIB_001:23;
then W1.reverse().append(W2) is_Walk_from x,y by A3,GLIB_001:31;
hence ex W being Walk of G st W is_Walk_from x,y;
end;
hence thesis;
end;
Lm8: for G being _Graph holds G is connected implies for v being Vertex of G
holds G.reachableFrom(v) = the_Vertices_of G
proof
let G be _Graph;
assume
A1: G is connected;
let v be Vertex of G;
now
let x be object;
thus x in G.reachableFrom(v) implies x in the_Vertices_of G;
assume x in the_Vertices_of G;
then ex W being Walk of G st W is_Walk_from v,x by A1;
hence x in G.reachableFrom(v) by Def5;
end;
hence thesis by TARSKI:2;
end;
Lm9: for G1,G2 being _Graph,v1 being Vertex of G1, v2 being Vertex of G2 st G1
== G2 & v1 = v2 holds G1.reachableFrom(v1) = G2.reachableFrom(v2)
proof
let G1,G2 be _Graph, v1 be Vertex of G1, v2 be Vertex of G2;
assume that
A1: G1 == G2 and
A2: v1 = v2;
now
let x be object;
hereby
assume x in G1.reachableFrom(v1);
then consider W being Walk of G1 such that
A3: W is_Walk_from v2,x by A2,Def5;
reconsider W2 = W as Walk of G2 by A1,GLIB_001:179;
W2 is_Walk_from v2,x by A3,GLIB_001:19;
hence x in G2.reachableFrom(v2) by Def5;
end;
assume x in G2.reachableFrom(v2);
then consider W being Walk of G2 such that
A4: W is_Walk_from v1,x by A2,Def5;
reconsider W2 = W as Walk of G1 by A1,GLIB_001:179;
W2 is_Walk_from v1,x by A4,GLIB_001:19;
hence x in G1.reachableFrom(v1) by Def5;
end;
hence thesis by TARSKI:2;
end;
Lm10: for G1 being _Graph, G2 being connected Subgraph of G1 holds G2 is
spanning implies G1 is connected
proof
let G1 be _Graph, G2 be connected Subgraph of G1;
assume
A1: G2 is spanning;
now
let u9,v9 be Vertex of G1;
reconsider u=u9, v=v9 as Vertex of G2 by A1,GLIB_000:def 33;
consider W being Walk of G2 such that
A2: W is_Walk_from u,v by Def1;
reconsider W as Walk of G1 by GLIB_001:167;
take W;
thus W is_Walk_from u9,v9 by A2,GLIB_001:19;
end;
hence thesis;
end;
Lm11: for G being _Graph holds G is connected iff G.componentSet() = {
the_Vertices_of G}
proof
let G be _Graph;
hereby
assume
A1: G is connected;
now
set v = the Vertex of G;
let x be object;
hereby
assume x in G.componentSet();
then ex v being Vertex of G st x = G.reachableFrom(v) by Def8;
then x = the_Vertices_of G by A1,Lm8;
hence x in {the_Vertices_of G} by TARSKI:def 1;
end;
assume x in {the_Vertices_of G};
then
A2: x = the_Vertices_of G by TARSKI:def 1;
G.reachableFrom(v) = the_Vertices_of G by A1,Lm8;
hence x in G.componentSet() by A2,Def8;
end;
hence G.componentSet() = {the_Vertices_of G} by TARSKI:2;
end;
assume G.componentSet() = {the_Vertices_of G};
then the_Vertices_of G in G.componentSet() by TARSKI:def 1;
then ex v being Vertex of G st G.reachableFrom(v) = the_Vertices_of G by Def8
;
hence thesis by Lm7;
end;
Lm12: for G1,G2 being _Graph holds G1 == G2 implies G1.componentSet() = G2
.componentSet()
proof
let G1,G2 be _Graph;
assume
A1: G1 == G2;
now
let x be object;
hereby
assume x in G1.componentSet();
then consider v1 being Vertex of G1 such that
A2: x = G1.reachableFrom(v1) by Def8;
reconsider v2 = v1 as Vertex of G2 by A1,GLIB_000:def 34;
x = G2.reachableFrom(v2) by A1,A2,Lm9;
hence x in G2.componentSet() by Def8;
end;
assume x in G2.componentSet();
then consider v2 being Vertex of G2 such that
A3: x = G2.reachableFrom(v2) by Def8;
reconsider v1 = v2 as Vertex of G1 by A1,GLIB_000:def 34;
x = G1.reachableFrom(v1) by A1,A3,Lm9;
hence x in G1.componentSet() by Def8;
end;
hence thesis by TARSKI:2;
end;
Lm13: for G being _Graph, x being set holds x in G.componentSet() implies x is
non empty Subset of the_Vertices_of G
proof
let G be _Graph, x be set;
assume x in G.componentSet();
then ex v being Vertex of G st x = G.reachableFrom(v) by Def8;
hence thesis;
end;
Lm14: for G being _Graph, C being Component of G holds the_Edges_of C = G
.edgesBetween(the_Vertices_of C)
proof
let G be _Graph, C be Component of G;
reconsider VC = the_Vertices_of C as Subset of the_Vertices_of G;
set EB = G.edgesBetween(VC);
C.edgesBetween(the_Vertices_of C) c= EB by GLIB_000:76;
then
A1: the_Edges_of C c= EB by GLIB_000:34;
now
let e be object;
thus e in the_Edges_of C implies e in EB by A1;
assume
A2: e in EB;
now
set GX = the inducedSubgraph of G,VC,EB;
assume
A3: not e in the_Edges_of C;
A4: the_Edges_of GX = EB by GLIB_000:def 37;
the_Vertices_of GX = VC by GLIB_000:def 37;
then
A5: C is spanning Subgraph of GX by A1,A4,GLIB_000:44,def 33;
then GX is connected by Lm10;
then not C c< GX by Def7;
then GX == C or not C c= GX by GLIB_000:def 36;
hence contradiction by A2,A3,A4,A5,GLIB_000:def 34,def 35;
end;
hence e in the_Edges_of C;
end;
hence thesis by TARSKI:2;
end;
Lm15: for G being _Graph, C1,C2 being Component of G holds the_Vertices_of C1
= the_Vertices_of C2 iff C1 == C2
proof
let G be _Graph, C1,C2 be Component of G;
hereby
assume
A1: the_Vertices_of C1 = the_Vertices_of C2;
then the_Edges_of C1 = G.edgesBetween(the_Vertices_of C2) by Lm14
.= the_Edges_of C2 by Lm14;
hence C1 == C2 by A1,GLIB_000:86;
end;
assume C1 == C2;
hence thesis by GLIB_000:def 34;
end;
Lm16: for G being _Graph, C being Component of G, v being Vertex of G holds v
in the_Vertices_of C iff the_Vertices_of C = G.reachableFrom(v)
proof
let G be _Graph, C be Component of G, v be Vertex of G;
hereby
assume
A1: v in the_Vertices_of C;
now
let x be object;
hereby
assume x in the_Vertices_of C;
then reconsider x9=x as Vertex of C;
consider W being Walk of C such that
A2: W is_Walk_from v,x9 by A1,Def1;
reconsider W as Walk of G by GLIB_001:167;
W is_Walk_from v,x by A2,GLIB_001:19;
hence x in G.reachableFrom(v) by Def5;
end;
assume
A3: x in G.reachableFrom(v);
then reconsider x9=x as Vertex of G;
A4: G.reachableFrom(x9) = G.reachableFrom(v) by A3,Lm3;
set CX = the inducedSubgraph of G,G.reachableFrom(x9);
not C c< CX by Def7;
then
A5: C == CX or not C c= CX by GLIB_000:def 36;
A6: the_Edges_of CX = G.edgesBetween(G.reachableFrom(x9)) by GLIB_000:def 37;
now
let e be object;
set v1 = (the_Source_of C).e, v2 = (the_Target_of C).e;
assume
A7: e in the_Edges_of C;
then reconsider v1,v2 as Vertex of C by FUNCT_2:5;
e Joins v1,v2,C by A7,GLIB_000:def 13;
then
A8: e Joins v1,v2,G by GLIB_000:72;
consider W1 being Walk of C such that
A9: W1 is_Walk_from v,v1 by A1,Def1;
reconsider W1 as Walk of G by GLIB_001:167;
consider W2 being Walk of C such that
A10: W2 is_Walk_from v,v2 by A1,Def1;
reconsider W2 as Walk of G by GLIB_001:167;
W2 is_Walk_from v,v2 by A10,GLIB_001:19;
then
A11: v2 in G.reachableFrom(x9) by A4,Def5;
W1 is_Walk_from v,v1 by A9,GLIB_001:19;
then v1 in G.reachableFrom(x9) by A4,Def5;
hence e in the_Edges_of CX by A6,A8,A11,GLIB_000:32;
end;
then
A12: the_Edges_of C c= the_Edges_of CX;
A13: the_Vertices_of CX = G.reachableFrom(x9) by GLIB_000:def 37;
now
let z be object;
assume z in the_Vertices_of C;
then consider W being Walk of C such that
A14: W is_Walk_from v,z by A1,Def1;
reconsider W as Walk of G by GLIB_001:167;
W is_Walk_from v,z by A14,GLIB_001:19;
hence z in the_Vertices_of CX by A13,A4,Def5;
end;
then the_Vertices_of C c= the_Vertices_of CX;
then
A15: C is Subgraph of CX by A12,GLIB_000:44;
x in the_Vertices_of CX by A13,Lm1;
hence x in the_Vertices_of C by A5,A15,GLIB_000:def 34,def 35;
end;
hence the_Vertices_of C = G.reachableFrom(v) by TARSKI:2;
end;
assume the_Vertices_of C = G.reachableFrom(v);
hence thesis by Lm1;
end;
Lm17: for G being _Graph, C1,C2 being Component of G, v being set st v in
the_Vertices_of C1 & v in the_Vertices_of C2 holds C1 == C2
proof
let G be _Graph, C1,C2 be Component of G, v be set;
assume that
A1: v in the_Vertices_of C1 and
A2: v in the_Vertices_of C2;
reconsider v9 = v as Vertex of G by A1;
the_Vertices_of C1 = G.reachableFrom(v9) & the_Vertices_of C2 = G
.reachableFrom(v9) by A1,A2,Lm16;
hence thesis by Lm15;
end;
Lm18: for G being _Graph holds G is connected iff G.numComponents() = 1
proof
let G be _Graph;
hereby
assume G is connected;
then G.componentSet() = {the_Vertices_of G} by Lm11;
hence G.numComponents() = 1 by CARD_1:30;
end;
assume G.numComponents() = 1;
then consider V being object such that
A1: G.componentSet() = {V} by CARD_2:42;
reconsider V as set by TARSKI:1;
now
let v be object;
assume v in the_Vertices_of G;
then reconsider v9=v as Vertex of G;
now
set V2 = G.reachableFrom(v9);
assume
A2: not v in V;
V2 in G.componentSet() by Def8;
then not v in V2 by A1,A2,TARSKI:def 1;
hence contradiction by Lm1;
end;
hence v in V;
end;
then
A3: the_Vertices_of G c= V;
V in G.componentSet() by A1,TARSKI:def 1;
then V = the_Vertices_of G by A3,XBOOLE_0:def 10;
hence thesis by A1,Lm11;
end;
Lm19: for G being connected _Graph, v being Vertex of G holds v is non
cut-vertex iff for G2 being removeVertex of G,v holds G2.numComponents() c= G
.numComponents()
proof
let G be connected _Graph, v be Vertex of G;
hereby
assume v is non cut-vertex;
then consider G3 being removeVertex of G,v such that
A1: not G.numComponents() in G3.numComponents();
let G2 be removeVertex of G,v;
G3.numComponents() c= G.numComponents() by A1,CARD_1:4;
hence G2.numComponents() c= G.numComponents() by Lm12,GLIB_000:93;
end;
assume
A2: for G2 being removeVertex of G,v holds G2.numComponents() c= G
.numComponents();
now
set X = the removeVertex of G,v;
assume for G3 being removeVertex of G,v holds G.numComponents() in G3
.numComponents();
then
A3: G.numComponents() in X.numComponents();
X.numComponents() c= G.numComponents() by A2;
hence contradiction by A3,CARD_1:4;
end;
hence thesis;
end;
Lm20: for G being connected _Graph, v being Vertex of G, G2 being removeVertex
of G,v st not v is cut-vertex holds G2 is connected
proof
let G be connected _Graph, v be Vertex of G, G2 be removeVertex of G,v;
assume
A1: not v is cut-vertex;
G.numComponents() = 1 by Lm18;
then G2.numComponents() c= succ 0 by A1,Lm19;
then G2.numComponents() c= {} \/ {{}};
then G2.numComponents() = {} \/ {{}} by ZFMISC_1:33
.= succ 0
.= 1;
hence thesis by Lm18;
end;
Lm21: for G being non trivial finite connected _Graph holds ex v1,v2 being
Vertex of G st v1 <> v2 & not v1 is cut-vertex & not v2 is cut-vertex
proof
let G be non trivial finite connected _Graph;
defpred P[Nat] means for G being non trivial finite connected _Graph st G
.order() = $1 holds ex v1,v2 being Vertex of G st v1 <> v2 & not v1 is
cut-vertex & not v2 is cut-vertex;
now
let k be Nat;
assume
A1: for n being Nat st n < k holds P[n];
now
let G be non trivial finite connected _Graph such that
A2: G.order() = k;
A3: G.numComponents() = 1 by Lm18;
now
per cases;
suppose
A4: not ex v being Vertex of G st v is cut-vertex;
consider v1,v2 being Vertex of G such that
A5: v1 <> v2 by GLIB_000:21;
take v1,v2;
thus v1 <> v2 by A5;
thus not v1 is cut-vertex & not v2 is cut-vertex by A4;
end;
suppose
ex cv being Vertex of G st cv is cut-vertex;
then consider cv being Vertex of G such that
A6: cv is cut-vertex;
set G2 = the removeVertex of G,cv;
1 < G2.numComponents() by A3,A6;
then 1+1 <= G2.numComponents() by NAT_1:13;
then card 2 <= card G2.componentSet();
then Segm card 2 c= Segm card G2.componentSet() by NAT_1:39;
then 2 c= card G2.componentSet();
then consider C1,C2 being object such that
A7: C1 in G2.componentSet() & C2 in G2.componentSet() and
A8: C1 <> C2 by PENCIL_1:2;
reconsider C1,C2 as Element of G2.componentSet() by A7;
set CC1 = the inducedSubgraph of G2,C1;
set CC2 = the inducedSubgraph of G2,C2;
A9: the_Vertices_of G2 = (the_Vertices_of G) \ {cv} by GLIB_000:47;
G.edgesBetween( (the_Vertices_of G) \ {cv} ) = (the_Edges_of G)
\ (G.edgesInOut({cv})) by GLIB_000:35;
then
G.edgesBetween( (the_Vertices_of G) \ {cv} ) = (the_Edges_of G)
\ cv.edgesInOut() by GLIB_000:def 40;
then
A10: the_Edges_of G2 = (the_Edges_of G) \ cv.edgesInOut() by GLIB_000:47;
A11: G2.order() + 1 = k by A2,GLIB_000:48;
A12: now
let C be Component of G2;
now
set x = the Vertex of C;
assume
A13: for a being Vertex of C holds not ex e being set st e
Joins cv,a,G;
the_Vertices_of C c= the_Vertices_of G2;
then reconsider x9=x as Vertex of G2;
the_Vertices_of G2 c= the_Vertices_of G;
then reconsider x99=x9 as Vertex of G;
consider W being Walk of G such that
A14: W is_Walk_from cv,x99 by Def1;
set P = the Path of W;
A15: P is_Walk_from cv,x99 by A14,GLIB_001:160;
then
A16: P.first() = cv by GLIB_001:def 23;
set P2 = P.cut(2*1+1, len P);
A17: 1 <= len P by ABIAN:12;
A18: P.last() = x by A15,GLIB_001:def 23;
then
A19: P.(len P)=x by GLIB_001:def 7;
not x9 in {cv} by A9,XBOOLE_0:def 5;
then
A20: x <> cv by TARSKI:def 1;
then
A21: P is non trivial by A16,A18,GLIB_001:127;
then
A22: 2*1+1 <= len P by GLIB_001:125;
then P2 is_Walk_from P.3, P.(len P) by GLIB_001:37;
then
A23: P2 is_Walk_from P.3, x by A18,GLIB_001:def 7;
A24: P.(2*0+1)=cv by A16,GLIB_001:def 6;
now
assume cv in P2.vertices();
then consider n being odd Element of NAT such that
A25: n <= len P2 and
A26: P2.n = cv by GLIB_001:87;
A27: 1 <= n by ABIAN:12;
then
A28: 1+0 < n+2 by XREAL_1:8;
A29: n in dom P2 by A25,A27,FINSEQ_3:25;
then 3+n-1 in dom P by A22,GLIB_001:47;
then
A30: 2+n <= len P by FINSEQ_3:25;
P.(3+n-1) = cv by A22,A26,A29,GLIB_001:47;
hence contradiction by A20,A24,A19,A30,A28,GLIB_001:def 28;
end;
then reconsider P2 as Walk of G2 by GLIB_001:171;
P2 is_Walk_from P.3, x by A23,GLIB_001:19;
then P2.reverse() is_Walk_from x,P.3 by GLIB_001:23;
then
A31: P.3 in G2.reachableFrom(x9) by Def5;
len P <> 1 by A21,GLIB_001:125;
then 2*0+1 < len P by A17,XXREAL_0:1;
then P.(1+1) Joins P.1,P.(1+2),G by GLIB_001:def 3;
then
A32: P.2 Joins cv, P.3, G by A16,GLIB_001:def 6;
the_Vertices_of C = G2.reachableFrom(x9) by Lm16;
hence contradiction by A13,A32,A31;
end;
then consider a being Vertex of C, e being set such that
A33: e Joins cv,a,G;
A34: e in the_Edges_of G by A33,GLIB_000:def 13;
now
per cases;
suppose
C is trivial;
then consider v99 being Vertex of C such that
A35: the_Vertices_of C = {v99} by GLIB_000:22;
the_Vertices_of C c= the_Vertices_of G2;
then reconsider v9=v99 as Vertex of G2;
reconsider v=v9 as Vertex of G by GLIB_000:42;
take v;
thus v in the_Vertices_of C;
not v9 in {cv} by A9,XBOOLE_0:def 5;
then
A36: v9 <> cv by TARSKI:def 1;
A37: {v9} = G2.reachableFrom(v9) by A35,Lm16;
now
set G3 = the removeVertex of G,v;
A38: now
let e, z be set;
assume
A39: e Joins v,z,G;
then
A40: e in the_Edges_of G by GLIB_000:def 13;
now
assume that
A41: z <> v and
A42: z <> cv;
not e in cv.edgesInOut() by A36,A39,A42,GLIB_000:65;
then e in the_Edges_of G2 by A10,A40,XBOOLE_0:def 5;
then e Joins v,z,G2 by A39,GLIB_000:73;
then z in G2.reachableFrom(v9) by Lm1,Lm2;
hence contradiction by A37,A41,TARSKI:def 1;
end;
hence z = v or z = cv;
end;
now
let x,y be Vertex of G3;
now
per cases;
suppose
A43: x = y;
set W = G3.walkOf(x);
take W;
thus W is_Walk_from x,y by A43,GLIB_001:13;
end;
suppose
A44: x <> y;
reconsider x9=x,y9=y as Vertex of G by GLIB_000:42;
consider W being Walk of G such that
A45: W is_Walk_from x9,y9 by Def1;
set P = the Path of W;
A46: P is_Walk_from x9,y9 by A45,GLIB_001:160;
A47: the_Vertices_of G3 = (the_Vertices_of G)\{v} by
GLIB_000:47;
then not x in {v} by XBOOLE_0:def 5;
then x <> v by TARSKI:def 1;
then
A48: v <> P.1 by A46,GLIB_001:17;
not y in {v} by A47,XBOOLE_0:def 5;
then y <> v by TARSKI:def 1;
then
A49: v <> P.(len P) by A46,GLIB_001:17;
now
assume v in P.vertices();
then consider n being odd Element of NAT such that
A50: n <= len P and
A51: P.n = v by GLIB_001:87;
1 <= n by ABIAN:12;
then 1 < n by A48,A51,XXREAL_0:1;
then 1+1 <= n by NAT_1:13;
then reconsider n2 = n-2*1 as odd Element of NAT by
INT_1:5;
A52: now
A53: n2 < n - 0 by XREAL_1:15;
assume P.n2 = v;
hence contradiction by A48,A50,A51,A53,
GLIB_001:def 28;
end;
set e1 = P.(n2+1), e2 = P.(n+1);
n - 2 < len P - 0 by A50,XREAL_1:15;
then e1 Joins P.n2, P.(n2+2), G by GLIB_001:def 3;
then
A54: P.n2 = cv by A38,A51,A52,GLIB_000:14;
A55: n < len P by A49,A50,A51,XXREAL_0:1;
then
A56: n+2 <= len P by GLIB_001:1;
A57: now
A58: n+0 < n+2 by XREAL_1:8;
assume P.(n+2) = v;
hence contradiction by A48,A51,A56,A58,
GLIB_001:def 28;
end;
n2 < n - 0 by XREAL_1:15;
then
A59: n2+0 < n+2 by XREAL_1:8;
e2 Joins v, P.(n+2), G by A51,A55,GLIB_001:def 3;
then
A60: P.(n+2) = cv by A38,A57;
then cv = P.1 by A54,A56,A59,GLIB_001:def 28;
then
A61: cv = x by A46,GLIB_001:17;
cv = P.(len P) by A54,A56,A60,A59,GLIB_001:def 28;
hence contradiction by A44,A46,A61,GLIB_001:17;
end;
then reconsider P as Walk of G3 by GLIB_001:171;
take P;
thus P is_Walk_from x,y by A46,GLIB_001:19;
end;
end;
hence ex P being Walk of G3 st P is_Walk_from x,y;
end;
then
A62: G3 is connected;
assume v is cut-vertex;
then 1 < G3.numComponents() by A3;
hence contradiction by A62,Lm18;
end;
hence not v is cut-vertex;
end;
suppose
C is non trivial;
then reconsider C9=C as non trivial _Graph;
C.order()+0 < G2.order()+1 by GLIB_000:75,XREAL_1:8;
then consider v1, v2 being Vertex of C9 such that
A63: v1 <> v2 and
A64: not v1 is cut-vertex and
A65: not v2 is cut-vertex by A1,A11;
set C9R1 = the removeVertex of C9,v1;
A66: C9R1 is connected by A64,Lm20;
set C9R2 = the removeVertex of C9,v2;
A67: the_Vertices_of C9R1 = (the_Vertices_of C9) \ {v1} by
GLIB_000:47;
v2 in the_Vertices_of G2 by GLIB_000:42;
then not v2 in {cv} by A9,XBOOLE_0:def 5;
then
A68: v2 <> cv by TARSKI:def 1;
A69: the_Vertices_of C9R2 = (the_Vertices_of C9) \ {v2} by
GLIB_000:47;
v1 in the_Vertices_of G2 by GLIB_000:42;
then not v1 in {cv} by A9,XBOOLE_0:def 5;
then
A70: v1 <> cv by TARSKI:def 1;
A71: C9R2 is connected by A65,Lm20;
now
per cases;
suppose
A72: not v1 in cv.allNeighbors();
reconsider v9=v1 as Vertex of G2 by GLIB_000:42;
reconsider v =v9 as Vertex of G by GLIB_000:42;
take v;
thus v in the_Vertices_of C;
set G3 = the removeVertex of G,v;
A73: the_Vertices_of G3 = (the_Vertices_of G) \ {v} by
GLIB_000:47;
not cv in {v} by A70,TARSKI:def 1;
then reconsider cv9 = cv as Vertex of G3 by A73,
XBOOLE_0:def 5;
A74: v1 <> a by A33,A72,GLIB_000:71;
A75: the_Vertices_of C = G2.reachableFrom(v9) by Lm16;
now
let y be Vertex of G3;
now
per cases;
suppose
y = cv;
then G3.walkOf(y) is_Walk_from cv9,y by GLIB_001:13;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
suppose
A76: y <> cv;
now
per cases;
suppose
A77: y in the_Vertices_of C;
not a in {v1} by A74,TARSKI:def 1;
then
A78: a in the_Vertices_of C9R1 by A67,XBOOLE_0:def 5;
not y in {v1} by A73,XBOOLE_0:def 5;
then y in the_Vertices_of C9R1 by A67,A77,
XBOOLE_0:def 5;
then consider W being Walk of C9R1 such that
A79: W is_Walk_from y,a by A66,A78;
A80: W.1 = y & W.(len W) = a by A79,GLIB_001:17;
A81: now
assume v in W.vertices();
then not v in {v1} by A67,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
not e in v.edgesInOut() by A33,A70,A74,
GLIB_000:65;
then e in (the_Edges_of G)\v.edgesInOut() by A34,
XBOOLE_0:def 5;
then e in (the_Edges_of G)\G.edgesInOut({v}) by
GLIB_000:def 40;
then e in G.edgesBetween((the_Vertices_of G)\{v
}) by GLIB_000:35;
then e in the_Edges_of G3 by GLIB_000:47;
then e Joins cv,a,G3 by A33,GLIB_000:73;
then
A82: e Joins a,cv,G3 by GLIB_000:14;
reconsider W as Walk of C by GLIB_001:167;
reconsider W as Walk of G2 by GLIB_001:167;
reconsider W as Walk of G by GLIB_001:167;
not v in W.vertices() by A81,GLIB_001:98;
then reconsider W as Walk of G3 by GLIB_001:171;
W is_Walk_from y,a by A80,GLIB_001:17;
then W.addEdge(e) is_Walk_from y,cv by A82,
GLIB_001:66;
then W.addEdge(e).reverse() is_Walk_from cv,y
by GLIB_001:23;
hence
ex W being Walk of G3 st W is_Walk_from cv9 ,y;
end;
suppose
A83: not y in the_Vertices_of C;
reconsider y9=y as Vertex of G by GLIB_000:42;
consider W being Walk of G such that
A84: W is_Walk_from cv,y9 by Def1;
set P = the Path of W;
A85: P is_Walk_from cv,y9 by A84,GLIB_001:160;
then
A86: P.(len P) = y9 by GLIB_001:17;
A87: P.1 = cv by A85,GLIB_001:17;
now
assume v in P.vertices();
then consider
n being odd Element of NAT such that
A88: n <= len P and
A89: P.n = v by GLIB_001:87;
set P2 = P.cut(n,len P);
A90: P2 is_Walk_from v, y9 by A86,A88,A89,
GLIB_001:37;
1 <= n by ABIAN:12;
then
A91: 1 < n by A70,A87,A89,XXREAL_0:1;
now
assume cv in P2.vertices();
then consider
m being odd Element of NAT such
that
A92: m <= len P2 and
A93: P2.m = cv by GLIB_001:87;
1 <= m by ABIAN:12;
then
A94: m in dom P2 by A92,FINSEQ_3:25;
then
A95: P.(n+m-1) = cv by A88,A93,GLIB_001:47;
1+1 < n+m by A91,ABIAN:12,XREAL_1:8;
then 1+1-1 < n+m-1 by XREAL_1:14;
then
A96: 2*0+1 < n+m-1;
A97: n+m-1 in dom P by A88,A94,GLIB_001:47;
then n+m-1 <= len P by FINSEQ_3:25;
hence contradiction by A76,A87,A86,A95,A97
,A96,GLIB_001:def 28;
end;
then reconsider P2 as Walk of G2 by
GLIB_001:171;
P2 is_Walk_from v,y9 by A90,GLIB_001:19;
hence contradiction by A75,A83,Def5;
end;
then reconsider P as Walk of G3 by GLIB_001:171;
take P;
thus P is_Walk_from cv9,y by A85,GLIB_001:19;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
then G3 is connected by Lm6;
then G3.numComponents() = 1 by Lm18;
hence not v is cut-vertex by A3;
end;
suppose
A98: v1 in cv.allNeighbors() & not v2 in cv .allNeighbors();
reconsider v9=v2 as Vertex of G2 by GLIB_000:42;
reconsider v =v9 as Vertex of G by GLIB_000:42;
take v;
thus v in the_Vertices_of C;
set G3 = the removeVertex of G,v;
A99: the_Vertices_of G3 = (the_Vertices_of G) \ {v} by
GLIB_000:47;
not cv in {v} by A68,TARSKI:def 1;
then reconsider cv9 = cv as Vertex of G3 by A99,
XBOOLE_0:def 5;
A100: v2 <> a by A33,A98,GLIB_000:71;
A101: the_Vertices_of C = G2.reachableFrom(v9) by Lm16;
now
let y be Vertex of G3;
now
per cases;
suppose
y = cv;
then G3.walkOf(y) is_Walk_from cv9,y by GLIB_001:13;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
suppose
A102: y <> cv;
now
per cases;
suppose
A103: y in the_Vertices_of C;
not a in {v2} by A100,TARSKI:def 1;
then
A104: a in the_Vertices_of C9R2 by A69,XBOOLE_0:def 5;
not y in {v2} by A99,XBOOLE_0:def 5;
then y in the_Vertices_of C9R2 by A69,A103,
XBOOLE_0:def 5;
then consider W being Walk of C9R2 such that
A105: W is_Walk_from y,a by A71,A104;
A106: W.1 = y & W.(len W) = a by A105,GLIB_001:17;
A107: now
assume v in W.vertices();
then not v in {v2} by A69,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
not e in v.edgesInOut() by A33,A68,A100,
GLIB_000:65;
then e in (the_Edges_of G) \ v.edgesInOut() by
A34,XBOOLE_0:def 5;
then e in (the_Edges_of G)\G.edgesInOut({v}) by
GLIB_000:def 40;
then e in G.edgesBetween(the_Vertices_of G\{v})
by GLIB_000:35;
then e in the_Edges_of G3 by GLIB_000:47;
then e Joins cv,a,G3 by A33,GLIB_000:73;
then
A108: e Joins a,cv,G3 by GLIB_000:14;
reconsider W as Walk of C by GLIB_001:167;
reconsider W as Walk of G2 by GLIB_001:167;
reconsider W as Walk of G by GLIB_001:167;
not v in W.vertices() by A107,GLIB_001:98;
then reconsider W as Walk of G3 by GLIB_001:171;
W is_Walk_from y,a by A106,GLIB_001:17;
then W.addEdge(e) is_Walk_from y,cv by A108,
GLIB_001:66;
then W.addEdge(e).reverse() is_Walk_from cv,y
by GLIB_001:23;
hence
ex W being Walk of G3 st W is_Walk_from cv9 ,y;
end;
suppose
A109: not y in the_Vertices_of C;
reconsider y9=y as Vertex of G by GLIB_000:42;
consider W being Walk of G such that
A110: W is_Walk_from cv,y9 by Def1;
set P = the Path of W;
A111: P is_Walk_from cv,y9 by A110,GLIB_001:160;
then
A112: P.(len P) = y9 by GLIB_001:17;
A113: P.1 = cv by A111,GLIB_001:17;
now
assume v in P.vertices();
then consider
n being odd Element of NAT such that
A114: n <= len P and
A115: P.n = v by GLIB_001:87;
set P2 = P.cut(n,len P);
A116: P2 is_Walk_from v, y9 by A112,A114,A115,
GLIB_001:37;
1 <= n by ABIAN:12;
then
A117: 1 < n by A68,A113,A115,XXREAL_0:1;
now
assume cv in P2.vertices();
then consider
m being odd Element of NAT such
that
A118: m <= len P2 and
A119: P2.m = cv by GLIB_001:87;
1 <= m by ABIAN:12;
then
A120: m in dom P2 by A118,FINSEQ_3:25;
then
A121: P.(n+m-1) = cv by A114,A119,GLIB_001:47;
1+1 < n+m by A117,ABIAN:12,XREAL_1:8;
then 1+1-1 < n+m-1 by XREAL_1:14;
then
A122: 2*0+1 < n+m-1;
A123: n+m-1 in dom P by A114,A120,GLIB_001:47;
then n+m-1 <= len P by FINSEQ_3:25;
hence contradiction by A102,A113,A112,A121
,A123,A122,GLIB_001:def 28;
end;
then reconsider P2 as Walk of G2 by
GLIB_001:171;
P2 is_Walk_from v,y9 by A116,GLIB_001:19;
hence contradiction by A101,A109,Def5;
end;
then reconsider P as Walk of G3 by GLIB_001:171;
take P;
thus P is_Walk_from cv9,y by A111,GLIB_001:19;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
then G3 is connected by Lm6;
then G3.numComponents() = 1 by Lm18;
hence not v is cut-vertex by A3;
end;
suppose
v1 in cv.allNeighbors() & v2 in cv .allNeighbors();
then consider e being object such that
A124: e Joins cv,v2,G by GLIB_000:71;
reconsider v9=v1 as Vertex of G2 by GLIB_000:42;
set a = v2;
reconsider v =v9 as Vertex of G by GLIB_000:42;
take v;
thus v in the_Vertices_of C;
set G3 = the removeVertex of G,v;
A125: the_Vertices_of G3 = (the_Vertices_of G) \ {v} by
GLIB_000:47;
not cv in {v} by A70,TARSKI:def 1;
then reconsider cv9 = cv as Vertex of G3 by A125,
XBOOLE_0:def 5;
A126: the_Vertices_of C = G2.reachableFrom(v9) by Lm16;
A127: e in the_Edges_of G by A124,GLIB_000:def 13;
now
let y be Vertex of G3;
now
per cases;
suppose
y = cv;
then G3.walkOf(y) is_Walk_from cv9,y by GLIB_001:13;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
suppose
A128: y <> cv;
now
per cases;
suppose
A129: y in the_Vertices_of C;
not a in {v1} by A63,TARSKI:def 1;
then
A130: a in the_Vertices_of C9R1 by A67,XBOOLE_0:def 5;
not y in {v1} by A125,XBOOLE_0:def 5;
then y in the_Vertices_of C9R1 by A67,A129,
XBOOLE_0:def 5;
then consider W being Walk of C9R1 such that
A131: W is_Walk_from y,a by A66,A130;
A132: W.1 = y & W.(len W) = a by A131,GLIB_001:17;
A133: now
assume v in W.vertices();
then not v in {v1} by A67,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
not e in v.edgesInOut() by A63,A70,A124,
GLIB_000:65;
then e in (the_Edges_of G)\v.edgesInOut() by A127
,XBOOLE_0:def 5;
then e in (the_Edges_of G)\G.edgesInOut({v}) by
GLIB_000:def 40;
then e in G.edgesBetween(the_Vertices_of G\{v})
by GLIB_000:35;
then e in the_Edges_of G3 by GLIB_000:47;
then e Joins cv,a,G3 by A124,GLIB_000:73;
then
A134: e Joins a,cv,G3 by GLIB_000:14;
reconsider W as Walk of C by GLIB_001:167;
reconsider W as Walk of G2 by GLIB_001:167;
reconsider W as Walk of G by GLIB_001:167;
not v in W.vertices() by A133,GLIB_001:98;
then reconsider W as Walk of G3 by GLIB_001:171;
W is_Walk_from y,a by A132,GLIB_001:17;
then W.addEdge(e) is_Walk_from y,cv by A134,
GLIB_001:66;
then W.addEdge(e).reverse() is_Walk_from cv,y
by GLIB_001:23;
hence
ex W being Walk of G3 st W is_Walk_from cv9 ,y;
end;
suppose
A135: not y in the_Vertices_of C;
reconsider y9=y as Vertex of G by GLIB_000:42;
consider W being Walk of G such that
A136: W is_Walk_from cv,y9 by Def1;
set P = the Path of W;
A137: P is_Walk_from cv,y9 by A136,GLIB_001:160;
then
A138: P.(len P) = y9 by GLIB_001:17;
A139: P.1 = cv by A137,GLIB_001:17;
now
assume v in P.vertices();
then consider
n being odd Element of NAT such that
A140: n <= len P and
A141: P.n = v by GLIB_001:87;
set P2 = P.cut(n,len P);
A142: P2 is_Walk_from v, y9 by A138,A140,A141,
GLIB_001:37;
1 <= n by ABIAN:12;
then
A143: 1 < n by A70,A139,A141,XXREAL_0:1;
now
assume cv in P2.vertices();
then consider
m being odd Element of NAT such
that
A144: m <= len P2 and
A145: P2.m = cv by GLIB_001:87;
1 <= m by ABIAN:12;
then
A146: m in dom P2 by A144,FINSEQ_3:25;
then
A147: P.(n+m-1) = cv by A140,A145,GLIB_001:47;
1+1 < n+m by A143,ABIAN:12,XREAL_1:8;
then 1+1-1 < n+m-1 by XREAL_1:14;
then
A148: 2*0+1 < n+m-1;
A149: n+m-1 in dom P by A140,A146,GLIB_001:47;
then n+m-1 <= len P by FINSEQ_3:25;
hence contradiction by A128,A139,A138,A147
,A149,A148,GLIB_001:def 28;
end;
then reconsider P2 as Walk of G2 by
GLIB_001:171;
P2 is_Walk_from v,y9 by A142,GLIB_001:19;
hence contradiction by A126,A135,Def5;
end;
then reconsider P as Walk of G3 by GLIB_001:171;
take P;
thus P is_Walk_from cv9,y by A137,GLIB_001:19;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
end;
hence ex W being Walk of G3 st W is_Walk_from cv9,y;
end;
then G3 is connected by Lm6;
then G3.numComponents() = 1 by Lm18;
hence not v is cut-vertex by A3;
end;
end;
hence ex v being Vertex of G st v in the_Vertices_of C & not v
is cut-vertex;
end;
end;
hence ex v being Vertex of G st v in the_Vertices_of C & not v is
cut-vertex;
end;
then consider v1 being Vertex of G such that
A150: v1 in the_Vertices_of CC1 and
A151: not v1 is cut-vertex;
consider v2 being Vertex of G such that
A152: v2 in the_Vertices_of CC2 and
A153: not v2 is cut-vertex by A12;
take v1,v2;
now
C2 is non empty Subset of the_Vertices_of G2 by Lm13;
then
A154: the_Vertices_of CC2 = C2 by GLIB_000:def 37;
C1 is non empty Subset of the_Vertices_of G2 by Lm13;
then
A155: the_Vertices_of CC1 = C1 by GLIB_000:def 37;
assume v1 = v2;
then CC1 == CC2 by A150,A152,Lm17;
hence contradiction by A8,A155,A154,GLIB_000:def 34;
end;
hence v1 <> v2;
thus not v1 is cut-vertex & not v2 is cut-vertex by A151,A153;
end;
end;
hence
ex v1,v2 being Vertex of G st v1 <> v2 & not v1 is cut-vertex & not
v2 is cut-vertex;
end;
hence P[k];
end;
then
A156: for k being Nat st for n being Nat st n < k holds P[n] holds P[k];
A157: for k being Nat holds P[k] from NAT_1:sch 4(A156);
G.order() = G.order();
hence thesis by A157;
end;
registration
let G be non trivial finite connected _Graph;
cluster non cut-vertex for Vertex of G;
existence
proof
ex v1,v2 being Vertex of G st v1 <> v2 &( not v1 is cut-vertex)& not v2
is cut-vertex by Lm21;
hence thesis;
end;
end;
Lm22: for G being acyclic _Graph, W being Path of G, e being set st not e in W
.edges() & e in W.last().edgesInOut() holds W.addEdge(e) is Path-like
proof
let G be acyclic _Graph, W be Path of G, e be set;
assume that
A1: not e in W.edges() and
A2: e in W.last().edgesInOut();
set v = W.last().adj(e), W2 = W.addEdge(e);
A3: e Joins W.last(), W.last().adj(e), G by A2,GLIB_000:67;
then
A4: len W2 = len W + 2 by GLIB_001:64;
A5: W2 is Trail-like by A1,A2,GLIB_001:142;
now
per cases;
suppose
A6: W is trivial;
then
for n being odd Element of NAT st 1 < n & n <= len W holds W.n <> v
by GLIB_001:126;
hence thesis by A1,A3,A6,GLIB_001:150;
end;
suppose
A7: W is non trivial;
A8: now
let n be odd Element of NAT;
assume that
A9: 1 < n and
A10: n <= len W;
now
set W3 = W2.cut(n,len W2);
assume
A11: W.n = v;
A12: n <= len W2 by A4,A10,NAT_1:12;
then
A13: W3.first()=W2.n by GLIB_001:37;
now
assume W3 is trivial;
then len W3 = 1 by GLIB_001:126;
then 1 + n = len W2 + 1 by A12,GLIB_001:36;
then 2 + len W - len W <= len W - len W by A4,A10,XREAL_1:13;
then 2 <= 0;
hence contradiction;
end;
then consider W4 being Path of W3 such that
A14: W4 is non trivial by A5,GLIB_001:166;
W3.last()=W2.(len W2) by A12,GLIB_001:37;
then
A15: W3.last() = W2.last() by GLIB_001:def 7
.= v by A3,GLIB_001:63;
n in dom W by A9,A10,FINSEQ_3:25;
then W3.first() = v by A3,A11,A13,GLIB_001:65;
then W4 is_Walk_from v,v by A15,GLIB_001:def 32;
then W4 is closed by GLIB_001:119;
then W4 is Cycle-like by A14,GLIB_001:def 31;
hence contradiction by Def2;
end;
hence W.n <> v;
end;
W is not closed by A7,GLIB_001:def 31,Def2;
hence thesis by A1,A3,A8,GLIB_001:150;
end;
end;
hence thesis;
end;
Lm23: for G being non trivial finite acyclic _Graph st the_Edges_of G <> {}
holds ex v1,v2 being Vertex of G st v1 <> v2 & v1 is endvertex & v2 is
endvertex & v2 in G.reachableFrom(v1)
proof
let G be non trivial finite acyclic _Graph such that
A1: the_Edges_of G <> {};
defpred P[Nat] means ex W being Path of G st len W = $1;
set e = the Element of the_Edges_of G;
A2: now
let k be Nat;
assume P[k];
then consider P being Path of G such that
A3: len P = k;
2*len P.vertexSeq() <= 2*(G.order()+1) by GLIB_001:154,XREAL_1:64;
then k + 1 <= 2*(G.order() + 1) by A3,GLIB_001:def 14;
then k+1-1 <= 2*(G.order() + 1) - 0 by XREAL_1:13;
hence k <= 2*(G.order()+1);
end;
set src = (the_Source_of G).e, tar = (the_Target_of G).e;
e Joins src,tar,G by A1,GLIB_000:def 13;
then
A4: len G.walkOf(src,e,tar) = 3 by GLIB_001:14;
then
A5: ex k being Nat st P[k];
consider k being Nat such that
A6: P[k] & for n being Nat st P[n] holds n <= k from NAT_1:sch 6(A2,A5
);
consider W being Path of G such that
A7: len W = k and
A8: for n being Nat st P[n] holds n <= k by A6;
A9: len W.reverse() = k by A7,GLIB_001:21;
A10: 3 <= len W by A4,A7,A8;
then
A11: W is non trivial by GLIB_001:125;
A12: now
assume W.first() = W.last();
then W is closed by GLIB_001:def 24;
then W is Cycle-like by A11,GLIB_001:def 31;
hence contradiction by Def2;
end;
then
A13: W is open by GLIB_001:def 24;
A14: now
let W be Path of G;
assume that
A15: len W = k and
A16: W is open;
2+1 <= len W by A4,A8,A15;
then 2 < len W by NAT_1:13;
then reconsider lenW2 = len W - 2*1 as odd Element of NAT by INT_1:5;
A17: lenW2 + 2 = len W;
A18: W is non trivial by A7,A10,A15,GLIB_001:125;
now
W.last() in W.vertices() by GLIB_001:88;
then not W.last() is isolated by A18,GLIB_001:135;
then W.last().degree() <> 0 by GLIB_000:def 50;
then card W.last().edgesInOut() <> 0 by GLIB_000:19;
then 0 < card W.last().edgesInOut() by NAT_1:3;
then
A19: 0+1 <= card W.last().edgesInOut() by NAT_1:13;
assume not W.last() is endvertex;
then W.last().degree() <> 1 by GLIB_000:def 52;
then card W.last().edgesInOut() <> 1 by GLIB_000:19;
then 1 < card W.last().edgesInOut() by A19,XXREAL_0:1;
then consider e1,e2 being set such that
A20: e1 in W.last().edgesInOut() and
A21: e2 in W.last().edgesInOut() & e1 <> e2 by NAT_1:59;
now
per cases;
suppose
A22: e1 = W.(lenW2+1);
take e2;
thus e2 in W.last().edgesInOut() & e2 <> W.(lenW2+1) by A21,A22;
end;
suppose
A23: e1 <> W.(lenW2+1);
take e1;
thus e1 in W.last().edgesInOut() & e1 <> W.(lenW2+1) by A20,A23;
end;
end;
then consider e being set such that
A24: e in W.last().edgesInOut() and
A25: e <> W.(lenW2+1);
consider v being Vertex of G such that
A26: e Joins W.last(),v,G by A24,GLIB_000:64;
now
per cases;
suppose
v in W.vertices();
then consider n being odd Element of NAT such that
A27: n <= len W and
A28: W.n = v by GLIB_001:87;
set m = W.rfind(n);
set W2 = W.cut(m,len W);
A29: m <= len W by A27,GLIB_001:def 22;
then W2.last() = W.(len W) by GLIB_001:37;
then
A30: e Joins W2.last(),v,G by A26,GLIB_001:def 7;
W.m = v by A27,A28,GLIB_001:def 22;
then
A31: W2.first() = v by A29,GLIB_001:37;
A32: e in W2.last().edgesInOut() by A30,GLIB_000:62;
now
per cases;
suppose
A33: not e in W2.edges();
A34: W2.addEdge(e) is non trivial by A30,GLIB_001:132;
W2.addEdge(e).first() = v & W2.addEdge(e).last() = v by A31,A30,
GLIB_001:63;
then
A35: W2.addEdge(e) is closed by GLIB_001:def 24;
W2.addEdge(e) is Path-like by A32,A33,Lm22;
then W2.addEdge(e) is Cycle-like by A35,A34,GLIB_001:def 31;
hence contradiction by Def2;
end;
suppose
A36: e in W2.edges();
W2.edges() c= W.edges() by GLIB_001:106;
then consider a being even Element of NAT such that
A37: 1 <= a and
A38: a <= len W and
A39: W.a = e by A36,GLIB_001:99;
reconsider a1 = a - 1 as odd Element of NAT by A37,INT_1:5;
A40: a1 < len W-0 by A38,XREAL_1:15;
a < lenW2 + 2 by A38,XXREAL_0:1;
then a+1 <= lenW2+1+1 by NAT_1:13;
then a <= lenW2+1 by XREAL_1:6;
then
A41: a < lenW2+1 by A25,A39,XXREAL_0:1;
a1+1 = a;
then
A42: e Joins W.a1, W.(a1+2), G by A39,A40,GLIB_001:def 3;
now
per cases by A26,A42,GLIB_000:15;
suppose
W.last() = W.a1 & v = W.(a1+2);
then W.(len W) = W.a1 by GLIB_001:def 7;
hence contradiction by A16,A40,GLIB_001:147;
end;
suppose
A43: W.last() = W.(a1+2) & v = W.a1;
a1 < lenW2+1-1 by A41,XREAL_1:14;
then
A44: a1+2 < len W by A17,XREAL_1:8;
W.(len W) = W.(a1+2) by A43,GLIB_001:def 7;
hence contradiction by A16,A44,GLIB_001:147;
end;
end;
hence contradiction;
end;
end;
hence contradiction;
end;
suppose
A45: not v in W.vertices();
A46: len W.addEdge(e) = k + 2 by A15,A26,GLIB_001:64;
W.addEdge(e) is Path-like by A16,A26,A45,GLIB_001:151;
then k+2 <= k+0 by A8,A46;
hence contradiction by XREAL_1:6;
end;
end;
hence contradiction;
end;
hence W.last() is endvertex;
end;
W is_Walk_from W.first(),W.last() by GLIB_001:def 23;
then
A47: W.last() in G.reachableFrom(W.first()) by Def5;
W.reverse() is open by A13,GLIB_001:120;
then W.reverse().last() is endvertex by A14,A9;
then
A48: W.first() is endvertex by GLIB_001:22;
W.last() is endvertex by A7,A13,A14;
hence thesis by A12,A48,A47;
end;
Lm24: for G being non trivial finite Tree-like _Graph holds ex v1,v2 being
Vertex of G st v1 <> v2 & v1 is endvertex & v2 is endvertex
proof
let G be non trivial finite Tree-like _Graph;
consider v1, v2 being Vertex of G such that
A1: v1 <> v2 by GLIB_000:21;
consider W being Walk of G such that
A2: W is_Walk_from v1,v2 by Def1;
A3: now
assume len W = 1;
then
A4: W.last() = W.1 by GLIB_001:def 7
.= W.first() by GLIB_001:def 6;
W.first() = v1 by A2,GLIB_001:def 23;
hence contradiction by A1,A2,A4,GLIB_001:def 23;
end;
1 <= len W by ABIAN:12;
then 1 < len W by A3,XXREAL_0:1;
then 1+1 <= len W by NAT_1:13;
then 1+1 in dom W by FINSEQ_3:25;
then W.(2*1) in the_Edges_of G by GLIB_001:8;
then ex v1,v2 being Vertex of G st v1 <> v2 & v1 is endvertex & v2 is
endvertex & v2 in G.reachableFrom(v1) by Lm23;
hence thesis;
end;
registration
let G be non trivial finite Tree-like _Graph;
cluster endvertex for Vertex of G;
existence
proof
consider v1,v2 being Vertex of G such that
v1 <> v2 and
A1: v1 is endvertex and
v2 is endvertex by Lm24;
take v1;
thus thesis by A1;
end;
end;
registration
let G be non trivial finite Tree-like _Graph, v be endvertex Vertex of G;
cluster -> Tree-like for removeVertex of G,v;
coherence
by Lm5;
end;
definition
let GSq be GraphSeq;
attr GSq is connected means
:Def12:
for n being Nat holds GSq.n is connected;
attr GSq is acyclic means
:Def13:
for n being Nat holds GSq.n is acyclic;
attr GSq is Tree-like means
for n being Nat holds GSq.n is Tree-like;
end;
registration
cluster trivial -> connected for GraphSeq;
coherence;
cluster trivial loopless -> Tree-like for GraphSeq;
coherence;
cluster acyclic -> simple for GraphSeq;
coherence
proof
let GSq be GraphSeq;
assume
A1: GSq is acyclic;
now
let n be Nat;
reconsider G = GSq.n as acyclic _Graph by A1;
G is simple;
hence GSq. n is simple;
end;
hence thesis by GLIB_000:def 63;
end;
cluster Tree-like -> acyclic connected for GraphSeq;
coherence
proof
let GSq be GraphSeq;
assume
A2: GSq is Tree-like;
now
let n be Nat;
reconsider G = GSq.n as Tree-like _Graph by A2;
G is acyclic;
hence GSq.n is acyclic & GSq.n is connected;
end;
hence thesis;
end;
cluster acyclic connected -> Tree-like for GraphSeq;
coherence
proof
let GSq be GraphSeq;
assume
A3: GSq is acyclic & GSq is connected;
let n be Nat;
reconsider G = GSq.n as acyclic connected _Graph by A3;
G is Tree-like;
hence thesis;
end;
end;
registration
cluster halting finite Tree-like for GraphSeq;
existence
proof
set G = the finite Tree-like _Graph;
set F = NAT --> G;
A1: dom F = NAT by FUNCOP_1:13;
reconsider F as ManySortedSet of NAT;
now
let x be Nat;
x in NAT by ORDINAL1:def 12;
then F.x in rng F by A1,FUNCT_1:3;
then F.x in {G} by FUNCOP_1:8;
hence F.x is _Graph by TARSKI:def 1;
end;
then reconsider F as GraphSeq by GLIB_000:def 53;
F.(1+1) in rng F by A1,FUNCT_1:3;
then F.(1+1) in {G} by FUNCOP_1:8;
then
A2: F.(1+1) = G by TARSKI:def 1;
take F;
F.1 in rng F by A1,FUNCT_1:3;
then F.1 in {G} by FUNCOP_1:8;
then F.1 = G by TARSKI:def 1;
hence F is halting by A2,GLIB_000:def 54;
now
let x be Nat;
x in NAT by ORDINAL1:def 12;
then F.x in rng F by A1,FUNCT_1:3;
then F.x in {G} by FUNCOP_1:8;
hence F.x is finite & F.x is Tree-like by TARSKI:def 1;
end;
hence thesis by GLIB_000:def 57;
end;
end;
registration
let GSq be connected GraphSeq, n be Nat;
cluster GSq.n -> connected for _Graph;
coherence by Def12;
end;
registration
let GSq be acyclic GraphSeq, n be Nat;
cluster GSq.n -> acyclic for _Graph;
coherence by Def13;
end;
registration
let GSq be Tree-like GraphSeq, n be Nat;
cluster GSq.n -> Tree-likefor _Graph;
coherence;
end;
begin :: Theorems
reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;
::$CT
theorem Th1:
for G being non trivial connected _Graph, v being Vertex of G
holds not v is isolated
proof
let G be non trivial connected _Graph, v be Vertex of G;
consider v1,v2 being Vertex of G such that
A1: v1 <> v2 by GLIB_000:21;
now
per cases;
suppose
v1 = v;
hence ex u being Vertex of G st u <> v by A1;
end;
suppose
v1 <> v;
hence ex u being Vertex of G st u <> v;
end;
end;
then consider u being Vertex of G such that
A2: u <> v;
consider W being Walk of G such that
A3: W is_Walk_from u,v by Def1;
A4: W.first() = u by A3,GLIB_001:def 23;
A5: W.last() = v by A3,GLIB_001:def 23;
then v in W.vertices() by GLIB_001:88;
hence thesis by A2,A4,A5,GLIB_001:127,135;
end;
theorem Th2:
for G1 being non trivial _Graph, v being Vertex of G1, G2 being
removeVertex of G1,v st G2 is connected & ex e being set st e in v.edgesInOut()
& not e Joins v,v,G1 holds G1 is connected
proof
let G1 be non trivial _Graph, v be Vertex of G1, G2 be removeVertex of G1,v;
assume that
A1: G2 is connected and
A2: ex e being set st e in v.edgesInOut() & not e Joins v,v,G1;
A3: now
let x be Vertex of G1;
assume x <> v;
then not x in {v} by TARSKI:def 1;
then x in (the_Vertices_of G1) \ {v} by XBOOLE_0:def 5;
hence x in (the_Vertices_of G2) by GLIB_000:47;
end;
consider e being set such that
A4: e in v.edgesInOut() and
A5: not e Joins v,v,G1 by A2;
set v3 = v.adj(e);
v <> v3 by A4,A5,GLIB_000:67;
then reconsider v39=v3 as Vertex of G2 by A3;
A6: e Joins v,v3,G1 by A4,GLIB_000:67;
then
A7: e Joins v3,v,G1 by GLIB_000:14;
now
let v1, v2 be Vertex of G1;
now
per cases;
suppose
v1 <> v;
then reconsider v19=v1 as Vertex of G2 by A3;
now
per cases;
suppose
v2 <> v;
then reconsider v29=v2 as Vertex of G2 by A3;
consider W9 being Walk of G2 such that
A8: W9 is_Walk_from v19,v29 by A1;
reconsider W=W9 as Walk of G1 by GLIB_001:167;
W is_Walk_from v1,v2 by A8,GLIB_001:19;
hence ex W being Walk of G1 st W is_Walk_from v1,v2;
end;
suppose
A9: v2 = v;
consider W9 being Walk of G2 such that
A10: W9 is_Walk_from v19,v39 by A1;
reconsider W=W9 as Walk of G1 by GLIB_001:167;
W is_Walk_from v1,v3 by A10,GLIB_001:19;
then W.first() = v1 & W.last() = v3 by GLIB_001:def 23;
then W.addEdge(e) is_Walk_from v1, v2 by A7,A9,GLIB_001:63;
hence ex W being Walk of G1 st W is_Walk_from v1,v2;
end;
end;
hence ex W being Walk of G1 st W is_Walk_from v1,v2;
end;
suppose
A11: v1 = v;
now
per cases;
suppose
v2 <> v;
then reconsider v29=v2 as Vertex of G2 by A3;
set W1 = G1.walkOf(v1,e,v3);
consider W29 being Walk of G2 such that
A12: W29 is_Walk_from v39,v29 by A1;
reconsider W2=W29 as Walk of G1 by GLIB_001:167;
A13: W2 is_Walk_from v3, v2 by A12,GLIB_001:19;
take W = W1.append(W2);
W1 is_Walk_from v1, v3 by A6,A11,GLIB_001:15;
hence W is_Walk_from v1,v2 by A13,GLIB_001:31;
end;
suppose
A14: v2 = v;
take W = G1.walkOf(v);
thus W is_Walk_from v1,v2 by A11,A14,GLIB_001:13;
end;
end;
hence ex W being Walk of G1 st W is_Walk_from v1,v2;
end;
end;
hence ex W being Walk of G1 st W is_Walk_from v1,v2;
end;
hence thesis;
end;
theorem
for G1 being non trivial connected _Graph, v being Vertex of G1, G2
being removeVertex of G1,v st v is endvertex holds G2 is connected by Lm5;
theorem Th4:
for G1 being connected _Graph, W being Walk of G1, e being set,
G2 being removeEdge of G1,e st W is Cycle-like & e in W.edges() holds G2 is
connected
proof
let G1 be connected _Graph, W be Walk of G1, e be set, G2 be removeEdge of
G1,e;
assume that
A1: W is Cycle-like and
A2: e in W.edges();
reconsider v1 = (the_Source_of G1).e, v2 = (the_Target_of G1).e as Vertex of
G1 by A2,FUNCT_2:5;
e Joins v1,v2,G1 by A2,GLIB_000:def 13;
then consider X being Walk of G1 such that
A3: X is_Walk_from v1,v2 and
A4: not e in X.edges() by A1,A2,GLIB_001:157;
reconsider X2 = X as Walk of G2 by A4,GLIB_001:172;
A5: X2 is_Walk_from v1,v2 by A3,GLIB_001:19;
then
A6: X2.reverse() is_Walk_from v2,v1 by GLIB_001:23;
now
let u,v be Vertex of G2;
the_Vertices_of G2 c= the_Vertices_of G1;
then reconsider u9=u,v9=v as Vertex of G1;
consider C being Walk of G1 such that
A7: C is_Walk_from u9,v9 by Def1;
set P = the Path of C;
A8: P is_Walk_from u9,v9 by A7,GLIB_001:160;
then
A9: P.(len P) = v by GLIB_001:17;
A10: P.1 = u by A8,GLIB_001:17;
now
per cases;
suppose
e in P.edges();
then consider
a,b being Vertex of G1, m being odd Element of NAT such that
A11: m+2 <= len P and
A12: a = P.m and
A13: e = P.(m+1) and
A14: b = P.(m+2) and
A15: e Joins a,b,G1 by GLIB_001:103;
set P1 = P.cut(1,m), P2 = P.cut(m+2, len P);
A16: m+2-2 < len P - 0 by A11,XREAL_1:15;
then
A17: m+1 <= len P by NAT_1:13;
now
assume e in P1.edges();
then consider x being even Element of NAT such that
A18: 1 <= x and
A19: x <= len P1 and
A20: P1.x = e by GLIB_001:99;
x <= m by A16,A19,GLIB_001:45;
then
A21: x < m+1 by NAT_1:13;
x in dom P1 by A18,A19,FINSEQ_3:25;
then P.x = e by A16,A20,GLIB_001:46;
hence contradiction by A13,A17,A18,A21,GLIB_001:138;
end;
then reconsider P19=P1 as Walk of G2 by GLIB_001:172;
now
assume e in P2.edges();
then consider x being even Element of NAT such that
A22: 1 <= x and
A23: x <= len P2 and
A24: P2.x = e by GLIB_001:99;
reconsider x1 = x-1 as odd Element of NAT by A22,INT_1:5;
A25: x1 < len P2 - 0 by A23,XREAL_1:15;
then m+2+x1 in dom P by A11,GLIB_001:36;
then
A26: m+2+x1 <= len P by FINSEQ_3:25;
x1+1 = x;
then
A27: e = P.(m+2+x1) by A11,A24,A25,GLIB_001:36;
m+1 < m+1+1 by NAT_1:13;
then
A28: m+1+0 < m+2+x1 by NAT_1:2,XREAL_1:8;
1 <= m+1 by NAT_1:12;
hence contradiction by A13,A27,A26,A28,GLIB_001:138;
end;
then reconsider P29=P2 as Walk of G2 by GLIB_001:172;
reconsider a,b as Vertex of G2 by GLIB_000:51;
1 <= m by ABIAN:12;
then P1 is_Walk_from u,a by A10,A12,A16,GLIB_001:37,JORDAN12:2;
then
A29: P19 is_Walk_from u,a by GLIB_001:19;
P2 is_Walk_from b,v by A9,A11,A14,GLIB_001:37;
then
A30: P29 is_Walk_from b,v by GLIB_001:19;
now
per cases by A15,GLIB_000:def 13;
suppose
a = v1 & b = v2;
then P19.append(X2) is_Walk_from u,b by A5,A29,GLIB_001:31;
then P19.append(X2).append(P29) is_Walk_from u,v by A30,GLIB_001:31
;
hence ex W being Walk of G2 st W is_Walk_from u,v;
end;
suppose
b = v1 & a = v2;
then P19.append(X2.reverse()) is_Walk_from u,b by A6,A29,
GLIB_001:31;
then
P19.append(X2.reverse()).append(P29) is_Walk_from u,v by A30,
GLIB_001:31;
hence ex W being Walk of G2 st W is_Walk_from u,v;
end;
end;
hence ex W being Walk of G2 st W is_Walk_from u,v;
end;
suppose
not e in P.edges();
then reconsider P as Walk of G2 by GLIB_001:172;
take P;
thus P is_Walk_from u,v by A8,GLIB_001:19;
end;
end;
hence ex W being Walk of G2 st W is_Walk_from u,v;
end;
hence thesis;
end;
theorem
(ex v1 being Vertex of G st for v2 being Vertex of G ex W being Walk
of G st W is_Walk_from v1,v2) implies G is connected by Lm6;
theorem
for G being trivial _Graph holds G is connected;
theorem Th7:
G1 == G2 & G1 is connected implies G2 is connected
proof
assume that
A1: G1 == G2 and
A2: G1 is connected;
now
let u,v be Vertex of G2;
reconsider u9=u,v9=v as Vertex of G1 by A1,GLIB_000:def 34;
consider W9 being Walk of G1 such that
A3: W9 is_Walk_from u9,v9 by A2;
reconsider W = W9 as Walk of G2 by A1,GLIB_001:179;
take W;
thus W is_Walk_from u,v by A3,GLIB_001:19;
end;
hence thesis;
end;
theorem
v in G.reachableFrom(v) by Lm1;
theorem
for e,x,y being object holds
x in G.reachableFrom(v1) & e Joins x,y,G implies y in G.reachableFrom(
v1) by Lm2;
theorem
G.edgesBetween(G.reachableFrom(v)) = G.edgesInOut(G.reachableFrom(v))
proof
set R = G.reachableFrom(v);
now
let x be object;
set Sx = (the_Source_of G).x, Tx = (the_Target_of G).x;
assume
A1: x in G.edgesInOut(R);
then reconsider Sx, Tx as Vertex of G by FUNCT_2:5;
now
per cases by A1,GLIB_000:28;
suppose
A2: Sx in R;
then consider W being Walk of G such that
A3: W is_Walk_from v,Sx by Def5;
now
W.last() = Sx by A3,GLIB_001:def 23;
then
A4: x Joins W.last(), Tx,G by A1,GLIB_000:def 13;
assume
A5: not Tx in R;
W.first() = v by A3,GLIB_001:def 23;
then W.addEdge(x) is_Walk_from v, Tx by A4,GLIB_001:63;
hence contradiction by A5,Def5;
end;
then
A6: x in G.edgesInto(R) by A1,GLIB_000:def 26;
x in G.edgesOutOf(R) by A1,A2,GLIB_000:def 27;
then x in G.edgesInto(R) /\ G.edgesOutOf(R) by A6,XBOOLE_0:def 4;
hence x in G.edgesBetween(R) by GLIB_000:def 29;
end;
suppose
A7: Tx in R;
then consider W being Walk of G such that
A8: W is_Walk_from v,Tx by Def5;
now
W.last() = Tx by A8,GLIB_001:def 23;
then
A9: x Joins W.last(), Sx,G by A1,GLIB_000:def 13;
assume
A10: not Sx in R;
W.first() = v by A8,GLIB_001:def 23;
then W.addEdge(x) is_Walk_from v, Sx by A9,GLIB_001:63;
hence contradiction by A10,Def5;
end;
then
A11: x in G.edgesOutOf(R) by A1,GLIB_000:def 27;
x in G.edgesInto(R) by A1,A7,GLIB_000:def 26;
then x in G.edgesInto(R) /\ G.edgesOutOf(R) by A11,XBOOLE_0:def 4;
hence x in G.edgesBetween(R) by GLIB_000:def 29;
end;
end;
hence x in G.edgesBetween(R);
end;
then G.edgesBetween(R) c= G.edgesInOut(R) & G.edgesInOut(R) c= G
.edgesBetween(R) by GLIB_000:33;
hence thesis by XBOOLE_0:def 10;
end;
theorem
v1 in G.reachableFrom(v2) implies G.reachableFrom(v1) = G
.reachableFrom(v2) by Lm3;
theorem
v in W.vertices() implies W.vertices() c= G.reachableFrom(v) by Lm4;
theorem
for G1 being _Graph, G2 being Subgraph of G1, v1 being Vertex of G1,
v2 being Vertex of G2 st v1 = v2 holds G2.reachableFrom(v2) c= G1.reachableFrom
(v1)
proof
let G1 be _Graph, G2 be Subgraph of G1, v1 be Vertex of G1, v2 be Vertex of
G2;
assume
A1: v1 = v2;
let v be object;
assume v in G2.reachableFrom(v2);
then consider W being Walk of G2 such that
A2: W is_Walk_from v2,v by Def5;
reconsider W2=W as Walk of G1 by GLIB_001:167;
W2 is_Walk_from v1,v by A1,A2,GLIB_001:19;
hence thesis by Def5;
end;
theorem
(ex v being Vertex of G st G.reachableFrom(v) = the_Vertices_of G)
implies G is connected by Lm7;
theorem
G is connected implies for v being Vertex of G holds G.reachableFrom(v
) = the_Vertices_of G by Lm8;
theorem
for v1 being Vertex of G1, v2 being Vertex of G2 st G1 == G2 & v1 = v2
holds G1.reachableFrom(v1) = G2.reachableFrom(v2) by Lm9;
theorem
v in G.reachableDFrom(v)
proof
G.walkOf(v) is_Walk_from v,v by GLIB_001:13;
hence thesis by Def6;
end;
theorem
x in G.reachableDFrom(v1) & e DJoins x,y,G implies y in G .reachableDFrom(v1)
proof
set RFV = G.reachableDFrom(v1);
assume that
A1: x in RFV and
A2: e DJoins x,y,G;
consider W being directed Walk of G such that
A3: W is_Walk_from v1,x by A1,Def6;
W.addEdge(e) is directed & W.addEdge(e) is_Walk_from v1,y by A2,A3,
GLIB_001:123;
hence thesis by Def6;
end;
theorem
G.reachableDFrom(v) c= G.reachableFrom(v)
proof
set RFD = G.reachableDFrom(v);
let x be object;
assume
A1: x in RFD;
then reconsider x9=x as Vertex of G;
ex W being directed Walk of G st W is_Walk_from v,x9 by A1,Def6;
hence thesis by Def5;
end;
theorem
for G1 being _Graph, G2 being Subgraph of G1, v1 being Vertex of G1,
v2 being Vertex of G2 st v1 = v2 holds G2.reachableDFrom(v2) c= G1
.reachableDFrom(v1)
proof
let G1 be _Graph, G2 be Subgraph of G1, v1 be Vertex of G1, v2 be Vertex of
G2;
assume
A1: v1 = v2;
let v be object;
assume v in G2.reachableDFrom(v2);
then consider W being DWalk of G2 such that
A2: W is_Walk_from v2,v by Def6;
reconsider W as DWalk of G1 by GLIB_001:175;
W is_Walk_from v1,v by A1,A2,GLIB_001:19;
hence v in G1.reachableDFrom(v1) by Def6;
end;
theorem
for v1 being Vertex of G1, v2 being Vertex of G2 st G1 == G2 & v1 = v2
holds G1.reachableDFrom(v1) = G2.reachableDFrom(v2)
proof
let v1 be Vertex of G1, v2 be Vertex of G2;
assume that
A1: G1 == G2 and
A2: v1 = v2;
now
let x be object;
hereby
assume x in G1.reachableDFrom(v1);
then consider W being DWalk of G1 such that
A3: W is_Walk_from v2,x by A2,Def6;
reconsider W2 = W as DWalk of G2 by A1,GLIB_001:179,181;
W2 is_Walk_from v2,x by A3,GLIB_001:19;
hence x in G2.reachableDFrom(v2) by Def6;
end;
assume x in G2.reachableDFrom(v2);
then consider W being DWalk of G2 such that
A4: W is_Walk_from v1,x by A2,Def6;
reconsider W2 = W as DWalk of G1 by A1,GLIB_001:179,181;
W2 is_Walk_from v1,x by A4,GLIB_001:19;
hence x in G1.reachableDFrom(v1) by Def6;
end;
hence thesis by TARSKI:2;
end;
theorem
for G1 being _Graph, G2 being connected Subgraph of G1 holds G2 is
spanning implies G1 is connected by Lm10;
theorem
union G.componentSet() = the_Vertices_of G
proof
now
let x be object;
thus x in union G.componentSet() implies x in the_Vertices_of G;
assume x in the_Vertices_of G;
then reconsider x9=x as Vertex of G;
set Y = G.reachableFrom(x9);
x in Y & Y in G.componentSet() by Def8,Lm1;
hence x in union G.componentSet() by TARSKI:def 4;
end;
hence thesis by TARSKI:2;
end;
theorem
G is connected iff G.componentSet() = {the_Vertices_of G} by Lm11;
theorem
G1 == G2 implies G1.componentSet() = G2.componentSet() by Lm12;
theorem
x in G.componentSet() implies x is non empty Subset of the_Vertices_of
G by Lm13;
theorem
G is connected iff G.numComponents() = 1 by Lm18;
theorem
G1 == G2 implies G1.numComponents() = G2.numComponents() by Lm12;
theorem
G is Component of G iff G is connected
proof
thus G is Component of G implies G is connected;
A1: G is Subgraph of G by GLIB_000:40;
A2: now
given G2 being connected Subgraph of G such that
A3: G c< G2;
now
per cases by A1,A3,GLIB_000:98;
suppose
the_Vertices_of G c< the_Vertices_of G2;
hence contradiction by XBOOLE_0:def 8;
end;
suppose
the_Edges_of G c< the_Edges_of G2;
hence contradiction by XBOOLE_0:def 8;
end;
end;
hence contradiction;
end;
assume G is connected;
hence thesis by A2,Def7,GLIB_000:40;
end;
theorem
for C being Component of G holds the_Edges_of C = G.edgesBetween(
the_Vertices_of C) by Lm14;
theorem
for C1,C2 being Component of G holds the_Vertices_of C1 =
the_Vertices_of C2 iff C1 == C2 by Lm15;
theorem
for C being Component of G, v being Vertex of G holds v in
the_Vertices_of C iff the_Vertices_of C = G.reachableFrom(v) by Lm16;
theorem
for C1,C2 being Component of G, v being set st v in the_Vertices_of C1
& v in the_Vertices_of C2 holds C1 == C2 by Lm17;
theorem
for G being connected _Graph, v being Vertex of G holds v is non
cut-vertex iff for G2 being removeVertex of G,v holds G2.numComponents() c= G
.numComponents() by Lm19;
theorem
for G being connected _Graph, v being Vertex of G, G2 being
removeVertex of G,v st not v is cut-vertex holds G2 is connected by Lm20;
theorem
for G being non trivial finite connected _Graph holds ex v1,v2 being
Vertex of G st v1 <> v2 & not v1 is cut-vertex & not v2 is cut-vertex by Lm21;
theorem Th37:
v is cut-vertex implies G is non trivial
proof
assume
A1: v is cut-vertex;
now
assume G is trivial;
then reconsider G9=G as trivial _Graph;
set G2 = the removeVertex of G9,v;
G9.numComponents() = 1 & G2.numComponents() = 1 by Lm18;
then 1 in 1 by A1;
hence contradiction;
end;
hence thesis;
end;
theorem
for v1 being Vertex of G1, v2 being Vertex of G2 st G1 == G2 & v1 = v2
holds v1 is cut-vertex implies v2 is cut-vertex
proof
let v1 be Vertex of G1, v2 be Vertex of G2;
assume that
A1: G1 == G2 and
A2: v1 = v2 and
A3: v1 is cut-vertex;
A4: G1 is non trivial by A3,Th37;
then
A5: G2 is non trivial by A1,GLIB_000:89;
let G2A be removeVertex of G2,v2;
set G1A = the removeVertex of G1,v1;
G1.numComponents() = G2.numComponents() by A1,Lm12;
then
A6: G2.numComponents() in G1A.numComponents() by A3;
the_Vertices_of G1A = the_Vertices_of G1 \ {v2} by A2,A4,GLIB_000:47
.= the_Vertices_of G2 \ {v2} by A1,GLIB_000:def 34;
then
A7: the_Vertices_of G2A = the_Vertices_of G1A by A5,GLIB_000:47;
G2 is Subgraph of G1 by A1,GLIB_000:87;
then
A8: G2A is Subgraph of G1 by GLIB_000:43;
the_Edges_of G1A = G1.edgesBetween(the_Vertices_of G1 \ {v1}) by A4,
GLIB_000:47
.= G1.edgesBetween(the_Vertices_of G2 \ {v2}) by A1,A2,GLIB_000:def 34
.= G2.edgesBetween(the_Vertices_of G2 \ {v2}) by A1,GLIB_000:90;
then the_Edges_of G2A = the_Edges_of G1A by A5,GLIB_000:47;
hence thesis by A6,A7,A8,Lm12,GLIB_000:86;
end;
theorem Th39:
for G being finite connected _Graph holds G.order() <= G.size() + 1
proof
let G be finite connected _Graph;
defpred P[finite _Graph] means $1 is connected implies $1.order() <= $1
.size() + 1;
A1: now
let k be non zero Nat;
assume
A2: for Gk being finite _Graph st Gk.order() = k holds P[Gk];
let Gk1 be finite _Graph;
assume
A3: Gk1.order() = k+1;
now
A4: now
assume Gk1.order() = 1;
then k+1 = 0+1 by A3;
hence contradiction;
end;
assume Gk1 is connected;
then reconsider Gk19 = Gk1 as non trivial finite connected _Graph by A4,
GLIB_000:26;
consider v1,v2 being Vertex of Gk19 such that
v1 <> v2 and
A5: not v1 is cut-vertex and
not v2 is cut-vertex by Lm21;
set Gkb = the removeVertex of Gk19, v1;
A6: Gkb.order() + 1 = k + 1 & Gkb.size()+card v1.edgesInOut()=Gk1
.size() by A3,GLIB_000:48;
not v1 is isolated by Th1;
then v1.edgesInOut() <> {} by GLIB_000:def 49;
then 0 < card v1.edgesInOut() by NAT_1:3;
then
A7: 0+1 <= card v1.edgesInOut() by NAT_1:13;
Gkb is connected by A5,Lm20;
then k <= Gk1.size() - card v1.edgesInOut() + 1 by A2,A6;
then k+1 <= Gk1.size() + 1 - card v1.edgesInOut() + card v1
.edgesInOut() by A7,XREAL_1:7;
hence Gk1.order() <= Gk1.size() + 1 by A3;
end;
hence P[Gk1];
end;
A8: for G being finite _Graph st G.order() = 1 holds P[G] by NAT_1:12;
for G being finite _Graph holds P[G] from GLIB_000:sch 1(A8,A1);
hence thesis;
end;
theorem
for G being acyclic _Graph holds G is simple;
theorem
for G being acyclic _Graph, W being Path of G, e being set st not e in
W.edges() & e in W.last().edgesInOut() holds W.addEdge(e) is Path-like by Lm22;
theorem
for G being non trivial finite acyclic _Graph st the_Edges_of G <> {}
holds ex v1,v2 being Vertex of G st v1 <> v2 & v1 is endvertex & v2 is
endvertex & v2 in G.reachableFrom(v1) by Lm23;
theorem Th43:
G1 == G2 & G1 is acyclic implies G2 is acyclic
proof
assume that
A1: G1 == G2 and
A2: G1 is acyclic;
reconsider G19 = G1 as acyclic _Graph by A2;
G2 is Subgraph of G19 by A1,GLIB_000:87;
hence thesis;
end;
theorem
for G being non trivial finite Tree-like _Graph holds ex v1,v2 being
Vertex of G st v1 <> v2 & v1 is endvertex & v2 is endvertex by Lm24;
theorem Th45:
for G being finite _Graph holds G is Tree-like iff G is acyclic
& G.order() = G.size() + 1
proof
defpred P[Nat] means for G being finite acyclic _Graph st G.order() = $1 & G
.order() = G.size() + 1 holds G is connected;
let G be finite _Graph;
hereby
defpred P[Nat] means for T being finite Tree-like _Graph st T.order() = $1
holds $1 = T.size() + 1;
assume
A1: G is Tree-like;
hence G is acyclic;
now
let T be finite Tree-like _Graph;
set VT = the_Vertices_of T, ET = the_Edges_of T;
assume T.order() = 1;
then card VT = 1 by GLIB_000:def 24;
then consider v being object such that
A2: VT = {v} by CARD_2:42;
reconsider v as Vertex of T by A2,TARSKI:def 1;
now
given e being object such that
A3: e in ET;
(the_Target_of T).e in {v} by A2,A3,FUNCT_2:5;
then
A4: (the_Target_of T).e = v by TARSKI:def 1;
(the_Source_of T).e in {v} by A2,A3,FUNCT_2:5;
then (the_Source_of T).e = v by TARSKI:def 1;
then e Joins v,v,T by A3,A4,GLIB_000:def 13;
then T.walkOf(v,e,v) is Cycle-like by GLIB_001:156;
hence contradiction by Def2;
end;
then card ET = 0 by CARD_1:27,XBOOLE_0:def 1;
then T.size() = 0 by GLIB_000:def 25;
hence 1 = T.size() + 1;
end;
then
A5: P[1];
now
let k be non zero Nat;
assume
A6: for T being finite Tree-like _Graph st T.order() = k holds k =
T.size() + 1;
let T be finite Tree-like _Graph;
assume
A7: T.order() = k+1;
then T.order() <> 1 by XCMPLX_1:3;
then reconsider aT = T as non trivial finite Tree-like _Graph by
GLIB_000:26;
set v = the endvertex Vertex of aT;
set T2 = the removeVertex of aT,v;
T2.order() + 1 - 1 = k + 1 - 1 by A7,GLIB_000:48;
then
A8: k = T2.size() + 1 by A6;
card v.edgesInOut() = v.degree() by GLIB_000:19
.= 1 by GLIB_000:def 52;
hence k+1 = T.size() + 1 by A8,GLIB_000:48;
end;
then
A9: for k being non zero Nat st P[k] holds P[k+1];
for k being non zero Nat holds P[k] from NAT_1:sch 10(A5,A9);
hence G.order() = G.size() + 1 by A1;
end;
assume that
A10: G is acyclic and
A11: G.order() = G.size() + 1;
now
let k be non zero Element of NAT;
assume
A12: for G being finite acyclic _Graph st G.order() = k & G.order() =
G.size() + 1 holds G is connected;
let G be finite acyclic _Graph;
assume that
A13: G.order() = k+1 and
A14: G.order() = G.size()+1;
now
assume G.order() = 1;
then 0 + 1 = k + 1 by A13;
hence contradiction;
end;
then reconsider aG = G as non trivial finite acyclic _Graph by GLIB_000:26;
the_Edges_of G <> {} by A13,A14,CARD_1:27,GLIB_000:def 25;
then consider v,v2 being Vertex of aG such that
v <> v2 and
A15: v is endvertex and
v2 is endvertex and
v2 in aG.reachableFrom(v) by Lm23;
set G2 = the removeVertex of G,v;
A16: G2.order() + 1 = aG.order() & G2.size() + card v.edgesInOut() = aG
.size() by GLIB_000:48;
card v.edgesInOut() = v.degree() by GLIB_000:19
.= 1 by A15,GLIB_000:def 52;
then
A17: G2 is connected by A12,A13,A14,A16;
consider e being object such that
A18: v.edgesInOut() = {e} and
A19: not e Joins v,v,G by A15,GLIB_000:def 51;
e in v.edgesInOut() by A18,TARSKI:def 1;
hence G is connected by A17,A19,Th2;
end;
then
A20: for k being non zero Nat st P[k] holds P[k+1];
now
let G be finite acyclic _Graph;
assume that
A21: G.order() = 1 and
G.order() = G.size() + 1;
consider v being Vertex of G such that
A22: the_Vertices_of G = {v} by A21,GLIB_000:27;
now
let v1,v2 be Vertex of G;
v1 = v & v2 = v by A22,TARSKI:def 1;
then G.walkOf(v) is_Walk_from v1,v2 by GLIB_001:13;
hence ex W being Walk of G st W is_Walk_from v1,v2;
end;
hence G is connected;
end;
then
A23: P[1];
for k being non zero Nat holds P[k] from NAT_1:sch 10(A23,A20);
then G is connected by A10,A11;
hence thesis by A10;
end;
theorem
for G being finite _Graph holds G is Tree-like iff G is connected & G
.order() = G.size() + 1
proof
let G be finite _Graph;
thus G is Tree-like implies G is connected & G.order() = G.size()+1 by Th45;
assume that
A1: G is connected and
A2: G.order() = G.size() + 1;
now
assume not G is acyclic;
then consider W being Walk of G such that
A3: W is Cycle-like;
set e = the Element of W.edges();
set G2 = the removeEdge of G,e;
A4: W.edges() <> {} by A3,GLIB_001:136;
then e in W.edges();
then
A5: G2.order()=G.order() & G2.size()+1=G.size() by GLIB_000:52;
G2 is connected by A1,A3,A4,Th4;
then G2.size() + 1 + 1 <= G2.size() + 1 + 0 by A2,A5,Th39;
hence contradiction by XREAL_1:6;
end;
hence thesis by A1;
end;
theorem Th47:
G1 == G2 & G1 is Tree-like implies G2 is Tree-like
by Th7,Th43;
theorem
G is_DTree_rooted_at x implies x is Vertex of G
proof
set v = the Vertex of G;
assume G is_DTree_rooted_at x;
then ex W being DWalk of G st W is_Walk_from x,v;
hence thesis by GLIB_001:18;
end;
theorem
G1 == G2 & G1 is_DTree_rooted_at x implies G2 is_DTree_rooted_at x
proof
assume that
A1: G1 == G2 and
A2: G1 is_DTree_rooted_at x;
A3: now
let y be Vertex of G2;
reconsider y9 = y as Vertex of G1 by A1,GLIB_000:def 34;
consider W being DWalk of G1 such that
A4: W is_Walk_from x,y9 by A2;
reconsider W9=W as DWalk of G2 by A1,GLIB_001:179,181;
take W9;
thus W9 is_Walk_from x,y by A4,GLIB_001:19;
end;
G1 is Tree-like by A2;
then G2 is Tree-like by A1,Th47;
hence thesis by A3;
end;