SGRAPH1: The Formalisation of Simple Graphs

:: The Formalisation of Simple Graphs
:: by Yozo Toda
::
:: Received September 8, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

:: interval is defined as subset of reals in measure5.
:: so we use a symbol nat_interval here.
:: following the definition of Seg, I add an assumption 1 <= m.
:: but it is unnatural, I think.
:: I changed the proof of existence
:: so that the assumption (1 <= m) is not necessary.
:: now nat_interval has the very natural definition, I think (-:

definition

let m, n be Nat;

func nat_interval (m,n) -> Subset of NAT equals :: SGRAPH1:def 1
{ i where i is Nat : ( m <= i & i <= n ) } ;

proof end;

end;

:: deftheorem defines nat_interval SGRAPH1:def 1 :

notation

let m, n be Nat;
synonym Seg (m,n) for nat_interval (m,n);

end;

registration

let m, n be Nat;

cluster nat_interval (m,n) -> finite ;

proof end;

end;

theorem :: SGRAPH1:1

for m, n being Nat
for e being set holds
( e in nat_interval (m,n) iff ex i being Nat st
( e = i & m <= i & i <= n ) ) ;

theorem :: SGRAPH1:2

for m, n, k being Nat holds
( k in nat_interval (m,n) iff ( m <= k & k <= n ) )

proof end;

theorem :: SGRAPH1:3

for n being Nat holds nat_interval (1,n) = Seg n ;

theorem Th4: :: SGRAPH1:4

for m, n being Nat st 1 <= m holds
nat_interval (m,n) c= Seg n

proof end;

theorem Th5: :: SGRAPH1:5

for k, m, n being Nat st k < m holds
Seg k misses nat_interval (m,n)

proof end;

theorem :: SGRAPH1:6

for m, n being Nat st n < m holds
nat_interval (m,n) = {}

proof end;

Lm1: for A being set
for s being Subset of A
for n being set st n in A holds
s \/ {n} is Subset of A

proof end;

definition

let A be set ;

func TWOELEMENTSETS A -> set equals :: SGRAPH1:def 2
{ z where z is Subset of A : card z = 2 } ;

end;

:: deftheorem defines TWOELEMENTSETS SGRAPH1:def 2 :

theorem Th7: :: SGRAPH1:7

for A, e being set holds
( e in TWOELEMENTSETS A iff ex z being Subset of A st
( e = z & card z = 2 ) ) ;

theorem Th8: :: SGRAPH1:8

for A, e being set holds
( e in TWOELEMENTSETS A iff ( e is finite Subset of A & ex x, y being object st
( x in A & y in A & x <> y & e = {x,y} ) ) )

proof end;

theorem Th9: :: SGRAPH1:9

for A being set holds TWOELEMENTSETS A c= bool A

proof end;

theorem Th10: :: SGRAPH1:10

for A, e1, e2 being set st {e1,e2} in TWOELEMENTSETS A holds
( e1 in A & e2 in A & e1 <> e2 )

proof end;

theorem Th11: :: SGRAPH1:11

TWOELEMENTSETS {} = {}

proof end;

registration

cluster TWOELEMENTSETS {} -> empty ;

coherence by Th11;

end;

theorem :: SGRAPH1:12

for t, u being set st t c= u holds
TWOELEMENTSETS t c= TWOELEMENTSETS u

proof end;

theorem :: SGRAPH1:13

canceled;

theorem :: SGRAPH1:14

canceled;

theorem :: SGRAPH1:15

canceled;

::$CT 3

registration

let A be finite set ;

cluster TWOELEMENTSETS A -> finite ;

coherence by Th9, FINSET_1:1;

end;

registration

let A be non trivial set ;

cluster TWOELEMENTSETS A -> non empty ;

proof end;

end;

registration

let a be set ;

cluster TWOELEMENTSETS {a} -> empty ;

proof end;

end;

:: graph is defined as a pair of two sets, of vertices and of edges.
:: we treat only simple graphs;
:: edges are non-directed, there is no loop,
:: between two vertices is at most one edge.
:: we define the set of all graphs SIMPLEGRAPHS,
:: and later define some operations on graphs
:: (contraction, deletion, minor, etc.) as relations on SIMPLEGRAPHS.
:: Vertices and Edges are used in GRAPH_1. so we must use different names.
:: we restrict simple graphs as finite ones
:: to treat degree as a cardinal of a set of edges.

definition

attr c₁ is strict ;
struct SimpleGraphStruct -> 1-sorted ;
aggr SimpleGraphStruct(# carrier, SEdges #) -> SimpleGraphStruct ;
sel SEdges c₁ -> Subset of (TWOELEMENTSETS the carrier of c₁);

end;

:: SIMPLEGRAPHS is the set of all (simple) graphs,
:: which is the smallest set satisfying following three conditions:
:: (1) it contains <Empty,Empty>,
:: (2) if <V,E> is an element of SIMPLEGRAPHS and n is not a vertex of <V,E>,
:: then <(V U {n}),E> is also an element of SIMPLEGRAPHS,
:: (3) if <V,E> is an element of SIMPLEGRAPHS,
:: v1,v2 are different vertices of <V,E>,
:: and {v1,v2} is not an edge of <V,E>,
:: then <V,(E U {v1,v2})> is also an element of SIMPLEGRAPHS.

definition

let X be set ;

func SIMPLEGRAPHS X -> set equals :: SGRAPH1:def 3
{ SimpleGraphStruct(# v,e #) where v is finite Subset of X, e is finite Subset of (TWOELEMENTSETS v) : verum } ;

end;

:: deftheorem defines SIMPLEGRAPHS SGRAPH1:def 3 :

theorem Th16: :: SGRAPH1:16

for X being set holds SimpleGraphStruct(# {},({} (TWOELEMENTSETS {})) #) in SIMPLEGRAPHS X

proof end;

registration

let X be set ;

cluster SIMPLEGRAPHS X -> non empty ;

coherence by Th16;

end;

definition

let X be set ;

mode SimpleGraph of X -> strict SimpleGraphStruct means :Def4: :: SGRAPH1:def 4
it is Element of SIMPLEGRAPHS X;

proof end;

end;

:: deftheorem Def4 defines SimpleGraph SGRAPH1:def 4 :

theorem Th17: :: SGRAPH1:17

for X being set
for g being object holds
( g in SIMPLEGRAPHS X iff ex v being finite Subset of X ex e being finite Subset of (TWOELEMENTSETS v) st g = SimpleGraphStruct(# v,e #) ) ;

theorem Th18: :: SGRAPH1:18

for X being set
for g being SimpleGraph of X holds
( the carrier of g c= X & the SEdges of g c= TWOELEMENTSETS the carrier of g )

proof end;

theorem :: SGRAPH1:19

for X being set
for g being SimpleGraph of X
for e being set st e in the SEdges of g holds
ex v1, v2 being object st
( v1 in the carrier of g & v2 in the carrier of g & v1 <> v2 & e = {v1,v2} ) by Th8;

theorem :: SGRAPH1:20

for X being set
for g being SimpleGraph of X
for v1, v2 being set st {v1,v2} in the SEdges of g holds
( v1 in the carrier of g & v2 in the carrier of g & v1 <> v2 ) by Th10;

theorem Th21: :: SGRAPH1:21

for X being set
for g being SimpleGraph of X holds
( the carrier of g is finite Subset of X & the SEdges of g is finite Subset of (TWOELEMENTSETS the carrier of g) )

proof end;

:: SECTION 3: equality relation on SIMPLEGRAPHS.
:: two graphs are said to be "isomorphic" if
:: (1) there is bijective function (i.e. set-theoretic isomorphism)
:: between two sets of vertices,
:: (2) this iso. respects the correspondence of edges.

definition

let X be set ;
let G, G9 be SimpleGraph of X;

pred G is_isomorphic_to G9 means :: SGRAPH1:def 5
ex Fv being Function of the carrier of G, the carrier of G9 st
( Fv is bijective & ( for v1, v2 being Element of G holds
( {v1,v2} in the SEdges of G iff {(Fv . v1),(Fv . v2)} in the SEdges of G ) ) );

end;

:: deftheorem defines is_isomorphic_to SGRAPH1:def 5 :

:: here is the induction principle on SIMPLEGRAPHS(X).

scheme :: SGRAPH1:sch 1
IndSimpleGraphs0{ F₁() -> set , P₁[ set ] } :

for G being set st G in SIMPLEGRAPHS F₁() holds
P₁[G]

provided

A1: P₁[ SimpleGraphStruct(# {},({} (TWOELEMENTSETS {})) #)] and
A2: for g being SimpleGraph of F₁()
for v being set st g in SIMPLEGRAPHS F₁() & P₁[g] & v in F₁() & not v in the carrier of g holds
P₁[ SimpleGraphStruct(# ( the carrier of g \/ {v}),({} (TWOELEMENTSETS ( the carrier of g \/ {v}))) #)] and
A3: for g being SimpleGraph of F₁()
for e being set st P₁[g] & e in TWOELEMENTSETS the carrier of g & not e in the SEdges of g holds
ex sege being Subset of (TWOELEMENTSETS the carrier of g) st
( sege = the SEdges of g \/ {e} & P₁[ SimpleGraphStruct(# the carrier of g,sege #)] )

proof end;

:: Now we give a theorem characterising SIMPLEGRAPHS as
:: an inductively defined set. We need some lemmas.

theorem :: SGRAPH1:22

for X being set
for g being SimpleGraph of X holds
( g = SimpleGraphStruct(# {},({} (TWOELEMENTSETS {})) #) or ex v being set ex e being Subset of (TWOELEMENTSETS v) st
( not v is empty & g = SimpleGraphStruct(# v,e #) ) )

proof end;

theorem Th23: :: SGRAPH1:23

for X being set
for V being Subset of X
for E being Subset of (TWOELEMENTSETS V)
for n being set
for Evn being finite Subset of (TWOELEMENTSETS (V \/ {n})) st SimpleGraphStruct(# V,E #) in SIMPLEGRAPHS X & n in X holds
SimpleGraphStruct(# (V \/ {n}),Evn #) in SIMPLEGRAPHS X

proof end;

theorem Th24: :: SGRAPH1:24

for X being set
for V being Subset of X
for E being Subset of (TWOELEMENTSETS V)
for v1, v2 being set st v1 in V & v2 in V & v1 <> v2 & SimpleGraphStruct(# V,E #) in SIMPLEGRAPHS X holds
ex v1v2 being finite Subset of (TWOELEMENTSETS V) st
( v1v2 = E \/ {{v1,v2}} & SimpleGraphStruct(# V,v1v2 #) in SIMPLEGRAPHS X )

proof end;

:: next we define a predicate
:: which describe how SIMPLEGRAPHS are generated inductively.
:: *** QUESTION ***
:: conditions (not n in V) and (not {v1,v2} in E) are redundant?

definition

let X, GG be set ;

pred GG is_SetOfSimpleGraphs_of X means :: SGRAPH1:def 6
( SimpleGraphStruct(# {},({} (TWOELEMENTSETS {})) #) in GG & ( for V being Subset of X
for E being Subset of (TWOELEMENTSETS V)
for n being set
for Evn being finite Subset of (TWOELEMENTSETS (V \/ {n})) st SimpleGraphStruct(# V,E #) in GG & n in X & not n in V holds
SimpleGraphStruct(# (V \/ {n}),Evn #) in GG ) & ( for V being Subset of X
for E being Subset of (TWOELEMENTSETS V)
for v1, v2 being set st SimpleGraphStruct(# V,E #) in GG & v1 in V & v2 in V & v1 <> v2 & not {v1,v2} in E holds
ex v1v2 being finite Subset of (TWOELEMENTSETS V) st
( v1v2 = E \/ {{v1,v2}} & SimpleGraphStruct(# V,v1v2 #) in GG ) ) );

end;

:: deftheorem defines is_SetOfSimpleGraphs_of SGRAPH1:def 6 :

theorem Th25: :: SGRAPH1:25

for X being set holds SIMPLEGRAPHS X is_SetOfSimpleGraphs_of X by Th16, Th23, Th24;

theorem Th26: :: SGRAPH1:26

for X, A being set st A is_SetOfSimpleGraphs_of X holds
SIMPLEGRAPHS X c= A

proof end;

theorem :: SGRAPH1:27

for X being set holds
( SIMPLEGRAPHS X is_SetOfSimpleGraphs_of X & ( for A being set st A is_SetOfSimpleGraphs_of X holds
SIMPLEGRAPHS X c= A ) ) by Th25, Th26;

:: graph G is a subgraph of graph G' if
:: (1) the set of vertices of G is a subset of the set of vertices of G',
:: (2) the set of edges of G is a subset of the set of edges of G',
:: where two endpoints of each edge of G must be the vertices of G.
:: (of course G must be a graph!)
:: now no lemma is proved )-:

definition

let X be set ;
let G be SimpleGraph of X;

mode SubGraph of G -> SimpleGraph of X means :: SGRAPH1:def 7
( the carrier of it c= the carrier of G & the SEdges of it c= the SEdges of G );

end;

:: deftheorem defines SubGraph SGRAPH1:def 7 :

:: the degree of a vertex means the number of edges connected to that vertex.
:: in the case of simple graphs, we can prove that
:: the degree is equal to the number of adjacent vertices.
:: (if loop is allowed,
:: the number of edges and the number of adjacent vertices are different.)
:: at first we defined degree(v),
:: where v was Element of the SEdges of(G) and G was an implicit argument.
:: but now we have changed the type of v to set,
:: and G must be an explicit argument
:: or we get an error "Inaccessible locus".

definition

let X be set ;
let G be SimpleGraph of X;
let v be set ;

func degree (G,v) -> Element of NAT means :Def8: :: SGRAPH1:def 8
ex X being finite set st
( ( for z being set holds
( z in X iff ( z in the SEdges of G & v in z ) ) ) & it = card X );

proof end;

proof end;

end;

:: deftheorem Def8 defines degree SGRAPH1:def 8 :

theorem Th28: :: SGRAPH1:28

for X being non empty set
for G being SimpleGraph of X
for v being set ex ww being finite set st
( ww = { w where w is Element of X : ( w in the carrier of G & {v,w} in the SEdges of G ) } & degree (G,v) = card ww )

proof end;

theorem :: SGRAPH1:29

for X being non empty set
for g being SimpleGraph of X
for v being set st v in the carrier of g holds
ex VV being finite set st
( VV = the carrier of g & degree (g,v) < card VV )

proof end;

theorem :: SGRAPH1:30

for X being set
for g being SimpleGraph of X
for v, e being set st e in the SEdges of g & degree (g,v) = 0 holds
not v in e

proof end;

theorem :: SGRAPH1:31

for X being set
for g being SimpleGraph of X
for v being set
for vg being finite set st vg = the carrier of g & v in vg & 1 + (degree (g,v)) = card vg holds
for w being Element of vg st v <> w holds
ex e being set st
( e in the SEdges of g & e = {v,w} )

proof end;

:: path is coded as a sequence of vertices,
:: any two of them contained are different each other.
:: but the head and the tail may be equal (which is cycle).

definition

let X be set ;
let g be SimpleGraph of X;
let v1, v2 be Element of g;
let p be FinSequence of the carrier of g;

pred p is_path_of v1,v2 means :: SGRAPH1:def 9
( p . 1 = v1 & p . (len p) = v2 & ( for i being Element of NAT st 1 <= i & i < len p holds
{(p . i),(p . (i + 1))} in the SEdges of g ) & ( for i, j being Element of NAT st 1 <= i & i < len p & i < j & j < len p holds
( p . i <> p . j & {(p . i),(p . (i + 1))} <> {(p . j),(p . (j + 1))} ) ) );

end;

:: deftheorem defines is_path_of SGRAPH1:def 9 :

definition

let X be set ;
let g be SimpleGraph of X;
let v1, v2 be Element of the carrier of g;

func PATHS (v1,v2) -> Subset of ( the carrier of g *) equals :: SGRAPH1:def 10
{ ss where ss is Element of the carrier of g * : ss is_path_of v1,v2 } ;

proof end;

end;

:: deftheorem defines PATHS SGRAPH1:def 10 :

theorem :: SGRAPH1:32

for X being set
for g being SimpleGraph of X
for v1, v2 being Element of the carrier of g
for e being set holds
( e in PATHS (v1,v2) iff ex ss being Element of the carrier of g * st
( e = ss & ss is_path_of v1,v2 ) ) ;

theorem :: SGRAPH1:33

for X being set
for g being SimpleGraph of X
for v1, v2 being Element of the carrier of g
for e being Element of the carrier of g * st e is_path_of v1,v2 holds
e in PATHS (v1,v2) ;

::definition :: is_cycle
:: let g be SimpleGraph of X,
:: v1,v2 be Element of (the carrier of g),
:: p be Element of PATHS(v1,v2);
:: pred p is_cycle means :cycleDef:
:: v1=v2 & ex q being Element of ((the carrier of g)*) st (q=p & 3<=(len q));
::end;

definition

let X be set ;
let g be SimpleGraph of X;
let p be set ;

pred p is_cycle_of g means :: SGRAPH1:def 11
ex v being Element of the carrier of g st p in PATHS (v,v);

end;

:: deftheorem defines is_cycle_of SGRAPH1:def 11 :

:: K_{3,3} = {{1,2,3,4,5,6},
:: {{1,4},{1,5},{1,6},{2,4},{2,5},{2,6},{3,4},{3,5},{3,6}}}.
:: K_5 = {{1,2,3,4,5},
:: {{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}}}.
:: for the proof of Kuratowski's theorem, we need only K_{3,3} and K_5.
:: here we define complete (and complete bipartate) graphs in general.

definition

let n, m be Element of NAT ;

func K_ (m,n) -> SimpleGraph of NAT means :: SGRAPH1:def 12
ex ee being Subset of (TWOELEMENTSETS (Seg (m + n))) st
( ee = { {i,j} where i, j is Element of NAT : ( i in Seg m & j in nat_interval ((m + 1),(m + n)) ) } & it = SimpleGraphStruct(# (Seg (m + n)),ee #) );

proof end;

end;

:: deftheorem defines K_ SGRAPH1:def 12 :

definition

let n be Element of NAT ;

func K_ n -> SimpleGraph of NAT means :Def13: :: SGRAPH1:def 13
ex ee being finite Subset of (TWOELEMENTSETS (Seg n)) st
( ee = { {i,j} where i, j is Element of NAT : ( i in Seg n & j in Seg n & i <> j ) } & it = SimpleGraphStruct(# (Seg n),ee #) );

proof end;

end;

:: deftheorem Def13 defines K_ SGRAPH1:def 13 :

:: TriangleGraph will be used in the definition of planegraphs.

definition

func TriangleGraph -> SimpleGraph of NAT equals :: SGRAPH1:def 14
K_ 3;

end;

:: deftheorem defines TriangleGraph SGRAPH1:def 14 :

theorem Th34: :: SGRAPH1:34

ex ee being Subset of (TWOELEMENTSETS (Seg 3)) st
( ee = {.{.1,2.},{.2,3.},{.3,1.}.} & TriangleGraph = SimpleGraphStruct(# (Seg 3),ee #) )

proof end;

theorem :: SGRAPH1:35

( the carrier of TriangleGraph = Seg 3 & the SEdges of TriangleGraph = {.{.1,2.},{.2,3.},{.3,1.}.} ) by Th34;

theorem :: SGRAPH1:36

( {1,2} in the SEdges of TriangleGraph & {2,3} in the SEdges of TriangleGraph & {3,1} in the SEdges of TriangleGraph ) by Th34, ENUMSET1:def 1;

theorem :: SGRAPH1:37

((<*1*> ^ <*2*>) ^ <*3*>) ^ <*1*> is_cycle_of TriangleGraph

proof end;