GRAPH_1: Graphs

:: Graphs
:: by Krzysztof Hryniewiecki
::
:: Received December 5, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct MultiGraphStruct -> 2-sorted ;
aggr MultiGraphStruct(# carrier, carrier', Source, Target #) -> MultiGraphStruct ;
sel Source c₁ -> Function of the carrier' of c₁, the carrier of c₁;
sel Target c₁ -> Function of the carrier' of c₁, the carrier of c₁;

end;

definition

let G be MultiGraphStruct ;
mode Vertex of G is Element of G;
mode Edge of G is Element of the carrier' of G;

end;

registration

cluster non empty non void strict for MultiGraphStruct ;

existence

proof end;

end;

definition

mode Graph is non empty MultiGraphStruct ;

end;

:: from CAT_1, 2008.07.02, A.T.

definition

let C be MultiGraphStruct ;
let f be Edge of C;

func dom f -> Vertex of C equals :Def1: :: GRAPH_1:def 1
the Source of C . f if ( not C is void & not C is empty )
otherwise the Vertex of C;

coherence

proof end;

consistency ;

func cod f -> Vertex of C equals :Def2: :: GRAPH_1:def 2
the Target of C . f if ( not C is void & not C is empty )
otherwise the Vertex of C;

correctness

proof end;

end;

:: deftheorem Def1 defines dom GRAPH_1:def 1 :

:: deftheorem Def2 defines cod GRAPH_1:def 2 :

definition

let C be non empty non void MultiGraphStruct ;
let f be Edge of C;

:: original: dom
redefine func dom f -> Vertex of C equals :: GRAPH_1:def 3
the Source of C . f;

compatibility by Def1;
coherence ;

:: original: cod
redefine func cod f -> Vertex of C equals :: GRAPH_1:def 4
the Target of C . f;

compatibility by Def2;
coherence ;

end;

:: deftheorem defines dom GRAPH_1:def 3 :

:: deftheorem defines cod GRAPH_1:def 4 :

definition

let G1, G2 be Graph;
assume that
A1: the Source of G1 tolerates the Source of G2 and
A2: the Target of G1 tolerates the Target of G2 ;

func G1 \/ G2 -> strict Graph means :Def5: :: GRAPH_1:def 5
( the carrier of it = the carrier of G1 \/ the carrier of G2 & the carrier' of it = the carrier' of G1 \/ the carrier' of G2 & ( for v being set st v in the carrier' 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 being set st v in the carrier' of G2 holds
( the Source of it . v = the Source of G2 . v & the Target of it . v = the Target of G2 . v ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines \/ GRAPH_1:def 5 :

definition

let G, G1, G2 be Graph;

pred G is_sum_of G1,G2 means :: GRAPH_1:def 6
( the Target of G1 tolerates the Target of G2 & the Source of G1 tolerates the Source of G2 & MultiGraphStruct(# the carrier of G, the carrier' of G, the Source of G, the Target of G #) = G1 \/ G2 );

end;

:: deftheorem defines is_sum_of GRAPH_1:def 6 :

definition

let IT be Graph;

attr IT is oriented means :Def7: :: GRAPH_1:def 7
for x, y being set st x in the carrier' of IT & y in the carrier' 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 :Def8: :: GRAPH_1:def 8
for x, y being set st x in the carrier' of IT & y in the carrier' 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 :Def9: :: GRAPH_1:def 9
for x being set holds
( not x in the carrier' of IT or not the Source of IT . x = the Target of IT . x );

attr IT is connected means :: GRAPH_1:def 10
for G1, G2 being Graph holds
( not the carrier of G1 misses the carrier of G2 or not IT is_sum_of G1,G2 );

end;

:: deftheorem Def7 defines oriented GRAPH_1:def 7 :

:: deftheorem Def8 defines non-multi GRAPH_1:def 8 :

:: deftheorem Def9 defines simple GRAPH_1:def 9 :

:: deftheorem defines connected GRAPH_1:def 10 :

definition

let IT be MultiGraphStruct ;

attr IT is finite means :Def11: :: GRAPH_1:def 11
( the carrier of IT is finite & the carrier' of IT is finite );

end;

:: deftheorem Def11 defines finite GRAPH_1:def 11 :

registration

cluster finite for MultiGraphStruct ;

existence

proof end;

cluster non empty V63() oriented non-multi simple connected finite for MultiGraphStruct ;

existence

proof end;

end;

registration

let G be finite MultiGraphStruct ;

cluster the carrier of G -> finite ;

coherence by Def11;

cluster the carrier' of G -> finite ;

coherence by Def11;

end;

definition

let G be Graph;
let x, y be Element of the carrier of G;
let v be set ;

pred v joins x,y means :: GRAPH_1:def 12
( ( 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;

:: deftheorem defines joins GRAPH_1:def 12 :

definition

let G be Graph;
let x, y be Element of the carrier of G;

pred x,y are_incident means :: GRAPH_1:def 13
ex v being set st
( v in the carrier' of G & v joins x,y );

end;

:: deftheorem defines are_incident GRAPH_1:def 13 :

definition

let G be Graph;

mode Chain of G -> FinSequence means :Def14: :: GRAPH_1:def 14
( ( for n being Nat st 1 <= n & n <= len it holds
it . n in the carrier' of G ) & ex p being FinSequence st
( len p = (len it) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
p . n in the carrier of G ) & ( for n being Nat st 1 <= n & n <= len it holds
ex x9, y9 being Vertex of G st
( x9 = p . n & y9 = p . (n + 1) & it . n joins x9,y9 ) ) ) );

existence

proof end;

end;

:: deftheorem Def14 defines Chain GRAPH_1:def 14 :

Lm1: for G being Graph holds {} is Chain of G

proof end;

definition

let G be Graph;
:: original: Chain
redefine mode Chain of G -> FinSequence of the carrier' of G;
coherence

proof end;

end;

definition

let G be Graph;
let IT be Chain of G;

attr IT is oriented means :: GRAPH_1:def 15
for n being Nat st 1 <= n & n < len IT holds
the Source of G . (IT . (n + 1)) = the Target of G . (IT . n);

end;

:: deftheorem defines oriented GRAPH_1:def 15 :

registration

let G be Graph;

cluster Relation-like NAT -defined the carrier' of G -valued Function-like empty finite FinSequence-like FinSubsequence-like for Chain of G;

existence

proof end;

end;

registration

let G be Graph;

cluster empty -> oriented for Chain of G;

coherence ;

end;

definition

let G be Graph;
let IT be Chain of G;

:: original: one-to-one
redefine attr IT is one-to-one means :: GRAPH_1:def 16
for n, m being Nat st 1 <= n & n < m & m <= len IT holds
IT . n <> IT . m;

compatibility

proof end;

end;

:: deftheorem defines one-to-one GRAPH_1:def 16 :

definition

let G be Graph;
mode Path of G is V7() Chain of G;

end;

definition

let G be Graph;
mode OrientedPath of G is V7() oriented Chain of G;

end;

definition

let G be Graph;
let IT be Path of G;

attr IT is cyclic means :: GRAPH_1:def 17
ex p being FinSequence st
( len p = (len IT) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
p . n in the carrier of G ) & ( for n being Nat st 1 <= n & n <= len IT holds
ex x9, y9 being Vertex of G st
( x9 = p . n & y9 = p . (n + 1) & IT . n joins x9,y9 ) ) & p . 1 = p . (len p) );

end;

:: deftheorem defines cyclic GRAPH_1:def 17 :

registration

let G be Graph;

cluster one-to-one empty -> cyclic for Chain of G;

coherence

proof end;

end;

definition

let G be Graph;
mode Cycle of G is cyclic Path of G;

end;

definition

let G be Graph;
mode OrientedCycle of G is cyclic OrientedPath of G;

end;

definition

let G be Graph;

mode Subgraph of G -> Graph means :Def18: :: GRAPH_1:def 18
( the carrier of it c= the carrier of G & the carrier' of it c= the carrier' of G & ( for v being set st v in the carrier' 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 carrier of it & the Target of G . v in the carrier of it ) ) );

existence

proof end;

end;

:: deftheorem Def18 defines Subgraph GRAPH_1:def 18 :

registration

let G be Graph;

cluster non empty V63() strict for Subgraph of G;

existence

proof end;

end;

definition

let G be finite Graph;

func VerticesCount G -> Element of NAT means :: GRAPH_1:def 19
ex B being finite set st
( B = the carrier of G & it = card B );

existence

proof end;

uniqueness ;

func EdgesCount G -> Element of NAT means :: GRAPH_1:def 20
ex B being finite set st
( B = the carrier' of G & it = card B );

existence

proof end;

uniqueness ;

end;

:: deftheorem defines VerticesCount GRAPH_1:def 19 :

:: deftheorem defines EdgesCount GRAPH_1:def 20 :

definition

let G be finite Graph;
let x be Vertex of G;

func EdgesIn x -> Element of NAT means :: GRAPH_1:def 21
ex X being finite set st
( ( for z being set holds
( z in X iff ( z in the carrier' of G & the Target of G . z = x ) ) ) & it = card X );

existence

proof end;

uniqueness

proof end;

func EdgesOut x -> Element of NAT means :: GRAPH_1:def 22
ex X being finite set st
( ( for z being set holds
( z in X iff ( z in the carrier' of G & the Source of G . z = x ) ) ) & it = card X );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines EdgesIn GRAPH_1:def 21 :

:: deftheorem defines EdgesOut GRAPH_1:def 22 :

definition

let G be finite Graph;
let x be Vertex of G;

func Degree x -> Element of NAT equals :: GRAPH_1:def 23
(EdgesIn x) + (EdgesOut x);

correctness ;

end;

:: deftheorem defines Degree GRAPH_1:def 23 :

Lm2: for n being Nat
for G being Graph
for p being Chain of G holds p | (Seg n) is Chain of G

proof end;

Lm3: for G being Graph
for H1, H2 being strict Subgraph of G st the carrier of H1 = the carrier of H2 & the carrier' of H1 = the carrier' of H2 holds
H1 = H2

proof end;

definition

let G1, G2 be Graph;

pred G1 c= G2 means :Def24: :: GRAPH_1:def 24
G1 is Subgraph of G2;

reflexivity

proof end;

end;

:: deftheorem Def24 defines c= GRAPH_1:def 24 :

Lm4: for G being Graph
for H being Subgraph of G holds
( the Source of H in PFuncs ( the carrier' of G, the carrier of G) & the Target of H in PFuncs ( the carrier' of G, the carrier of G) )

proof end;

definition

let G be Graph;

func bool G -> set means :Def25: :: GRAPH_1:def 25
for x being object holds
( x in it iff x is strict Subgraph of G );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def25 defines bool GRAPH_1:def 25 :

scheme :: GRAPH_1:sch 1
GraphSeparation{ F₁() -> Graph, P₁[ object ] } :

ex X being set st
for x being set holds
( x in X iff ( x is strict Subgraph of F₁() & P₁[x] ) )

proof end;

:: Properties of graphs ::

theorem :: GRAPH_1:1

for G being Graph holds
( dom the Source of G = the carrier' of G & dom the Target of G = the carrier' of G ) by FUNCT_2:def 1;

theorem :: GRAPH_1:2

for G being Graph
for x being Vertex of G holds x in the carrier of G ;

theorem :: GRAPH_1:3

for G being Graph
for v being set st v in the carrier' of G holds
( the Source of G . v in the carrier of G & the Target of G . v in the carrier of G ) by FUNCT_2:5;

:: Chain ::

theorem :: GRAPH_1:4

for n being Nat
for G being Graph
for p being Chain of G holds p | (Seg n) is Chain of G by Lm2;

:: Sum of two graphs ::

theorem Th5: :: GRAPH_1:5

for G, G1 being Graph st G1 c= G holds
( the Source of G1 c= the Source of G & the Target of G1 c= the Target of G )

proof end;

theorem :: GRAPH_1:6

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
( 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 )

proof end;

theorem Th7: :: GRAPH_1:7

for G being strict Graph holds G = G \/ G

proof end;

theorem Th8: :: GRAPH_1:8

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
G1 \/ G2 = G2 \/ G1

proof end;

theorem Th9: :: GRAPH_1:9

for G1, G2, G3 being Graph st 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 holds
(G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3)

proof end;

theorem Th10: :: GRAPH_1:10

for G, G1, G2 being Graph st G is_sum_of G1,G2 holds
G is_sum_of G2,G1 by Th8;

theorem :: GRAPH_1:11

for G being strict Graph holds G is_sum_of G,G by Th7;

:: Graphs' inclusion ::

theorem Th12: :: GRAPH_1:12

for G1, G2 being Graph st ex G being Graph st
( G1 c= G & G2 c= G ) holds
G1 \/ G2 = G2 \/ G1

proof end;

theorem :: GRAPH_1:13

for G1, G2, G3 being Graph st ex G being Graph st
( G1 c= G & G2 c= G & G3 c= G ) holds
(G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3)

proof end;

theorem :: GRAPH_1:14

for G being Graph holds {} is Chain of G by Lm1;

theorem :: GRAPH_1:15

for G being Graph
for H1, H2 being strict Subgraph of G st the carrier of H1 = the carrier of H2 & the carrier' of H1 = the carrier' of H2 holds
H1 = H2 by Lm3;

theorem Th16: :: GRAPH_1:16

for G1, G2 being strict Graph st G1 c= G2 & G2 c= G1 holds
G1 = G2

proof end;

theorem Th17: :: GRAPH_1:17

for G1, G2, G3 being Graph st G1 c= G2 & G2 c= G3 holds
G1 c= G3

proof end;

theorem Th18: :: GRAPH_1:18

for G, G1, G2 being Graph st G is_sum_of G1,G2 holds
( G1 c= G & G2 c= G )

proof end;

theorem Th19: :: GRAPH_1:19

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
( G1 c= G1 \/ G2 & G2 c= G1 \/ G2 )

proof end;

theorem Th20: :: GRAPH_1:20

for G1, G2 being Graph st ex G being Graph st
( G1 c= G & G2 c= G ) holds
( G1 c= G1 \/ G2 & G2 c= G1 \/ G2 )

proof end;

theorem Th21: :: GRAPH_1:21

for G, G1, G2, G3 being Graph st G1 c= G3 & G2 c= G3 & G is_sum_of G1,G2 holds
G c= G3

proof end;

theorem Th22: :: GRAPH_1:22

for G, G1, G2 being Graph st G1 c= G & G2 c= G holds
G1 \/ G2 c= G

proof end;

theorem :: GRAPH_1:23

for G1, G2 being strict Graph st G1 c= G2 holds
( G1 \/ G2 = G2 & G2 \/ G1 = G2 )

proof end;

theorem Th24: :: GRAPH_1:24

for G1, G2 being Graph st 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 ) holds
G1 c= G2

proof end;

theorem :: GRAPH_1:25

for G1 being Graph
for G being oriented Graph st G1 c= G holds
G1 is oriented

proof end;

theorem :: GRAPH_1:26

for G1 being Graph
for G being non-multi Graph st G1 c= G holds
G1 is non-multi

proof end;

theorem :: GRAPH_1:27

for G1 being Graph
for G being simple Graph st G1 c= G holds
G1 is simple

proof end;

:: Set of all subgraphs ::

theorem :: GRAPH_1:28

for G being Graph
for G1 being strict Graph holds
( G1 in bool G iff G1 c= G ) by Def25;

theorem Th29: :: GRAPH_1:29

for G being strict Graph holds G in bool G

proof end;

theorem :: GRAPH_1:30

for G2 being Graph
for G1 being strict Graph holds
( G1 c= G2 iff bool G1 c= bool G2 )

proof end;

theorem :: GRAPH_1:31

for G being strict Graph holds {G} c= bool G by Th29, ZFMISC_1:31;

theorem :: GRAPH_1:32

for G1, G2 being strict Graph st 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) & not G1 c= G2 holds
G2 c= G1

proof end;

theorem :: GRAPH_1:33

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
(bool G1) \/ (bool G2) c= bool (G1 \/ G2)

proof end;

theorem :: GRAPH_1:34

for G, G1, G2 being Graph st G1 in bool G & G2 in bool G holds
G1 \/ G2 in bool G

proof end;