Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Graphs

**by****Krzysztof Hryniewiecki**- Received December 5, 1990
- MML identifier: GRAPH_1

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, BOOLE, RELAT_1, PARTFUN1, RELAT_2, FINSET_1, ORDERS_1, FINSEQ_1, CARD_1, FUNCT_2, MCART_1, GRAPH_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSET_1, PARTFUN1, CARD_1, NAT_1, MCART_1; constructors FUNCT_2, FINSEQ_1, PARTFUN1, NAT_1, MCART_1, XREAL_0, MEMBERED, XBOOLE_0; clusters SUBSET_1, FINSEQ_1, FINSET_1, PARTFUN1, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve x, y, z, v for set, n, m, k for Nat; definition struct MultiGraphStruct (# Vertices, Edges -> set, Source, Target -> Function of the Edges, the Vertices #); end; definition let IT be MultiGraphStruct; attr IT is Graph-like means :: GRAPH_1:def 1 the Vertices of IT is non empty set; end; definition cluster strict Graph-like MultiGraphStruct; end; definition mode Graph is Graph-like MultiGraphStruct; end; reserve G, G1, G2, G3 for Graph; definition let G1, G2; assume (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2); func G1 \/ G2 -> strict Graph means :: GRAPH_1:def 2 the Vertices of it = (the Vertices of G1) \/ (the Vertices of G2) & the Edges of it = (the Edges of G1) \/ (the Edges of G2) & (for v st v in the Edges of G1 holds (the Source of it).v = (the Source of G1).v & (the Target of it).v = (the Target of G1).v) & (for v st v in the Edges of G2 holds (the Source of it).v = (the Source of G2).v & (the Target of it).v = (the Target of G2).v); end; definition let G, G1, G2 be Graph; pred G is_sum_of G1, G2 means :: GRAPH_1:def 3 (the Target of G1) tolerates (the Target of G2) & (the Source of G1) tolerates (the Source of G2) & the MultiGraphStruct of G = G1 \/ G2; end; definition let IT be Graph; attr IT is oriented means :: GRAPH_1:def 4 for x,y st x in the Edges of IT & y in the Edges of IT & (the Source of IT).x = (the Source of IT).y & (the Target of IT).x = (the Target of IT).y holds x = y; attr IT is non-multi means :: GRAPH_1:def 5 for x,y st x in the Edges of IT & y in the Edges of IT & (((the Source of IT).x = (the Source of IT).y & (the Target of IT).x = (the Target of IT).y) or ((the Source of IT).x = (the Target of IT).y & (the Source of IT).y = (the Target of IT).x)) holds x = y; attr IT is simple means :: GRAPH_1:def 6 not ex x st x in the Edges of IT & (the Source of IT).x = (the Target of IT).x; attr IT is connected means :: GRAPH_1:def 7 not ex G1, G2 being Graph st the Vertices of G1 misses the Vertices of G2 & IT is_sum_of G1, G2; end; definition let IT be MultiGraphStruct; attr IT is finite means :: GRAPH_1:def 8 the Vertices of IT is finite & the Edges of IT is finite; end; definition cluster finite MultiGraphStruct; cluster finite non-multi oriented simple connected Graph; end; reserve x, y for Element of (the Vertices of G); definition let G; let x, y; let v; pred v joins x, y means :: GRAPH_1:def 9 ((the Source of G).v = x & (the Target of G).v = y) or ((the Source of G).v = y & (the Target of G).v = x); end; definition let G; let x,y be Element of (the Vertices of G); pred x,y are_incydent means :: GRAPH_1:def 10 ex v being set st v in the Edges of G & v joins x, y; end; definition let G be Graph; mode Chain of G -> FinSequence means :: GRAPH_1:def 11 (for n st 1 <= n & n <= len it holds it.n in the Edges of G) & ex p being FinSequence st len p = len it + 1 & (for n st 1 <= n & n <= len p holds p.n in the Vertices of G) & for n st 1 <= n & n <= len it ex x', y' being Element of the Vertices of G st x' = p.n & y' = p.(n+1) & it.n joins x', y'; end; definition let G be Graph; redefine mode Chain of G -> FinSequence of the Edges of G; end; definition let G be Graph; let IT be Chain of G; attr IT is oriented means :: GRAPH_1:def 12 for n st 1 <= n & n < len IT holds (the Source of G).(IT.(n+1)) = (the Target of G).(IT.n); end; definition let G be Graph; cluster oriented Chain of G; end; definition let G be Graph; let IT be Chain of G; redefine attr IT is one-to-one means :: GRAPH_1:def 13 for n, m st 1 <= n & n < m & m <= len IT holds IT.n <> IT.m; end; definition let G be Graph; cluster one-to-one Chain of G; end; definition let G be Graph; mode Path of G is one-to-one Chain of G; end; definition let G be Graph; cluster one-to-one oriented Chain of G; end; definition let G be Graph; mode OrientedPath of G is one-to-one oriented Chain of G; end; definition let G be Graph; let IT be Path of G; canceled; attr IT is cyclic means :: GRAPH_1:def 15 ex p being FinSequence st len p = len IT + 1 & (for n st 1 <= n & n <= len p holds p.n in the Vertices of G) & (for n st 1 <= n & n <= len IT ex x', y' being Element of the Vertices of G st x' = p.n & y' = p.(n+1) & IT.n joins x', y') & p.1 = p.(len p); end; definition let G be Graph; cluster cyclic Path of G; end; definition let G be Graph; mode Cycle of G is cyclic Path of G; end; definition let G be Graph; cluster cyclic OrientedPath of G; end; definition let G be Graph; mode OrientedCycle of G is cyclic OrientedPath of G; end; definition let G be Graph; canceled; mode Subgraph of G -> Graph means :: GRAPH_1:def 17 the Vertices of it c= the Vertices of G & the Edges of it c= the Edges of G & for v st v in the Edges of it holds (the Source of it).v = (the Source of G).v & (the Target of it).v = (the Target of G).v & (the Source of G).v in the Vertices of it & (the Target of G).v in the Vertices of it; end; definition let G be Graph; cluster strict Subgraph of G; end; definition let G be finite Graph; func VerticesCount G -> Nat means :: GRAPH_1:def 18 ex B being finite set st B = the Vertices of G & it = card B; func EdgesCount G -> Nat means :: GRAPH_1:def 19 ex B being finite set st B = the Edges of G & it = card B; end; definition let G be finite Graph; let x be Element of the Vertices of G; func EdgesIn x -> Nat means :: GRAPH_1:def 20 ex X being finite set st (for z being set holds z in X iff z in the Edges of G & (the Target of G).z = x) & it = card(X); func EdgesOut x -> Nat means :: GRAPH_1:def 21 ex X being finite set st (for z being set holds z in X iff z in the Edges of G & (the Source of G).z = x) & it = card(X); end; definition let G be finite Graph; let x be Element of the Vertices of G; func Degree x -> Nat equals :: GRAPH_1:def 22 EdgesIn(x) + EdgesOut(x); end; definition let G1, G2 be Graph; pred G1 c= G2 means :: GRAPH_1:def 23 G1 is Subgraph of G2; reflexivity; end; definition let G be Graph; func bool G -> set means :: GRAPH_1:def 24 for x being set holds x in it iff x is strict Subgraph of G; end; scheme GraphSeparation{G() -> Graph, P[set]}: ex X being set st for x being set holds x in X iff x is strict Subgraph of G() & P[x]; :::::::::::::::::::::::::: :: Properties of graphs :: :::::::::::::::::::::::::: theorem :: GRAPH_1:1 for G being Graph holds dom(the Source of G) = the Edges of G & dom(the Target of G) = the Edges of G & rng(the Source of G) c= the Vertices of G & rng(the Target of G) c= the Vertices of G; theorem :: GRAPH_1:2 for x being Element of the Vertices of G holds x in the Vertices of G; theorem :: GRAPH_1:3 for v being set holds v in the Edges of G implies (the Source of G).v in the Vertices of G & (the Target of G).v in the Vertices of G; ::::::::::: :: Chain :: ::::::::::: theorem :: GRAPH_1:4 for p being Chain of G holds p|Seg(n) is Chain of G; ::::::::::::::::::::::: :: Sum of two graphs :: ::::::::::::::::::::::: theorem :: GRAPH_1:5 G1 c= G implies (the Source of G1) c= (the Source of G) & (the Target of G1) c= (the Target of G); theorem :: GRAPH_1:6 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) implies (the Source of (G1 \/ G2)) = (the Source of G1) \/ (the Source of G2) & (the Target of (G1 \/ G2)) = (the Target of G1) \/ (the Target of G2); theorem :: GRAPH_1:7 for G being strict Graph holds G = G \/ G; theorem :: GRAPH_1:8 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) implies G1 \/ G2 = G2 \/ G1; theorem :: GRAPH_1:9 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) & (the Source of G1) tolerates (the Source of G3) & (the Target of G1) tolerates (the Target of G3) & (the Source of G2) tolerates (the Source of G3) & (the Target of G2) tolerates (the Target of G3) implies (G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3); theorem :: GRAPH_1:10 G is_sum_of G1, G2 implies G is_sum_of G2, G1; theorem :: GRAPH_1:11 for G being strict Graph holds G is_sum_of G, G; ::::::::::::::::::::::: :: Graphs' inclusion :: ::::::::::::::::::::::: theorem :: GRAPH_1:12 (ex G st G1 c= G & G2 c= G) implies G1 \/ G2 = G2 \/ G1; theorem :: GRAPH_1:13 (ex G st G1 c= G & G2 c= G & G3 c= G) implies (G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3); theorem :: GRAPH_1:14 {} is cyclic oriented Path of G; theorem :: GRAPH_1:15 for H1, H2 being strict Subgraph of G st the Vertices of H1 = the Vertices of H2 & the Edges of H1 = the Edges of H2 holds H1 = H2; theorem :: GRAPH_1:16 for G1,G2 being strict Graph holds G1 c= G2 & G2 c= G1 implies G1 = G2; theorem :: GRAPH_1:17 G1 c= G2 & G2 c= G3 implies G1 c= G3; theorem :: GRAPH_1:18 G is_sum_of G1, G2 implies G1 c= G & G2 c= G; theorem :: GRAPH_1:19 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) implies G1 c= G1 \/ G2 & G2 c= G1 \/ G2; theorem :: GRAPH_1:20 (ex G st G1 c= G & G2 c= G) implies G1 c= G1 \/ G2 & G2 c= G1 \/ G2; theorem :: GRAPH_1:21 G1 c= G3 & G2 c= G3 & G is_sum_of G1, G2 implies G c= G3; theorem :: GRAPH_1:22 G1 c= G & G2 c= G implies (G1 \/ G2) c= G; theorem :: GRAPH_1:23 for G1,G2 being strict Graph holds G1 c= G2 implies G1 \/ G2 = G2 & G2 \/ G1 = G2; theorem :: GRAPH_1:24 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) & (G1 \/ G2 = G2 or G2 \/ G1 = G2) implies G1 c= G2; canceled 2; theorem :: GRAPH_1:27 for G being oriented Graph st G1 c= G holds G1 is oriented; theorem :: GRAPH_1:28 for G being non-multi Graph st G1 c= G holds G1 is non-multi; theorem :: GRAPH_1:29 for G being simple Graph st G1 c= G holds G1 is simple; :::::::::::::::::::::::::: :: Set of all subgraphs :: :::::::::::::::::::::::::: theorem :: GRAPH_1:30 for G1 being strict Graph holds G1 in bool G iff G1 c= G; theorem :: GRAPH_1:31 for G being strict Graph holds G in bool G; theorem :: GRAPH_1:32 for G1 being strict Graph holds G1 c= G2 iff bool G1 c= bool G2; canceled; theorem :: GRAPH_1:34 for G being strict Graph holds { G } c= bool G; theorem :: GRAPH_1:35 for G1,G2 being strict Graph holds (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) & bool (G1 \/ G2) c= (bool G1) \/ (bool G2) implies G1 c= G2 or G2 c= G1; theorem :: GRAPH_1:36 (the Source of G1) tolerates (the Source of G2) & (the Target of G1) tolerates (the Target of G2) implies bool G1 \/ bool G2 c= bool (G1 \/ G2); theorem :: GRAPH_1:37 G1 in bool G & G2 in bool G implies (G1 \/ G2) in bool G;

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