:: GLIB_002 semantic presentation

registration

let X be finite set ;
cluster bool X -> finite ;
coherence by FINSET_1:24;

end;

theorem Th1: :: GLIB_002:1

for X being finite set st 1 < card X holds
ex x1, x2 being set st
( x1 in X & x2 in X & x1 <> x2 )

proof end;

definition

let G be _Graph;
attr G is connected means :Def1: :: GLIB_002:def 1
for u, v being Vertex of G ex W being Walk of G st W is_Walk_from u,v;

end;

:: deftheorem Def1 defines connected GLIB_002:def 1 :

definition

let G be _Graph;
attr G is acyclic means :Def2: :: GLIB_002:def 2
for W being Walk of G holds not W is Cycle-like;

end;

:: deftheorem Def2 defines acyclic GLIB_002:def 2 :

definition

let G be _Graph;
attr G is Tree-like means :Def3: :: GLIB_002:def 3
( G is acyclic & G is connected );

end;

:: deftheorem Def3 defines Tree-like GLIB_002:def 3 :

registration

cluster trivial -> connected GraphStruct ;
coherence

proof end;

end;

registration

cluster loopless trivial -> Tree-like GraphStruct ;
coherence

proof end;

end;

registration

cluster acyclic -> simple GraphStruct ;
coherence

proof end;

end;

registration

cluster Tree-like -> simple connected acyclic GraphStruct ;
coherence by Def3;

end;

registration

cluster connected acyclic -> simple connected acyclic Tree-like GraphStruct ;
coherence by Def3;

end;

registration

let G be _Graph;
let v be Vertex of G;
cluster -> Tree-like inducedSubgraph of G,{v}, {} ;
coherence ;

end;

definition

let G be _Graph;
let v be set ;
pred G is_DTree_rooted_at v means :Def4: :: GLIB_002:def 4
( G is Tree-like & ( for x being Vertex of G ex W being DWalk of G st W is_Walk_from v,x ) );

end;

:: deftheorem Def4 defines is_DTree_rooted_at GLIB_002:def 4 :

registration

cluster finite trivial simple connected acyclic Tree-like GraphStruct ;
existence

proof end;

cluster finite non trivial simple connected acyclic Tree-like GraphStruct ;
existence

proof end;

end;

registration

let G be _Graph;
cluster finite trivial simple connected acyclic Tree-like Subgraph of G;
existence

proof end;

end;

registration

let G be acyclic _Graph;
cluster -> acyclic Subgraph of G;
coherence

proof end;

end;

definition

let G be _Graph;
let v be Vertex of G;
func G .reachableFrom v -> non empty Subset of (the_Vertices_of G) means :Def5: :: GLIB_002:def 5
for x being set holds
( x in it iff ex W being Walk of G st W is_Walk_from v,x );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines .reachableFrom GLIB_002:def 5 :

definition

let G be _Graph;
let v be Vertex of G;
func G .reachableDFrom v -> non empty Subset of (the_Vertices_of G) means :Def6: :: GLIB_002:def 6
for x being set holds
( x in it iff ex W being DWalk of G st W is_Walk_from v,x );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines .reachableDFrom GLIB_002:def 6 :

Lm1: for G being _Graph
for v being Vertex of G holds v in G .reachableFrom v

proof end;

Lm2: for G being _Graph
for v1 being Vertex of G
for e, x, y being set st x in G .reachableFrom v1 & e Joins x,y,G holds
y in G .reachableFrom v1

proof end;

Lm3: for G being _Graph
for v1, v2 being Vertex of G st v1 in G .reachableFrom v2 holds
G .reachableFrom v1 = G .reachableFrom v2

proof end;

Lm4: for G being _Graph
for W being Walk of G
for v being Vertex of G st v in W .vertices() holds
W .vertices() c= G .reachableFrom v

proof end;

definition

let G1 be _Graph;
let G2 be Subgraph of G1;
attr G2 is Component-like means :Def7: :: GLIB_002:def 7
( G2 is connected & ( for G3 being connected Subgraph of G1 holds not G2 c< G3 ) );

end;

:: deftheorem Def7 defines Component-like GLIB_002:def 7 :

registration

let G be _Graph;
cluster Component-like -> connected Subgraph of G;
coherence by Def7;

end;

registration

let G be _Graph;
let v be Vertex of G;
cluster -> connected Component-like inducedSubgraph of G,G .reachableFrom v,G .edgesBetween (G .reachableFrom v);
coherence

proof end;

end;

registration

let G be _Graph;
cluster connected Component-like Subgraph of G;
existence

proof 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: :: GLIB_002:def 8
for x being set holds
( x in it iff ex v being Vertex of G st x = G .reachableFrom v );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines .componentSet() GLIB_002:def 8 :

registration

let G be _Graph;
let X be Element of G .componentSet() ;
cluster -> connected Component-like inducedSubgraph of G,X,G .edgesBetween X;
coherence

proof end;

end;

definition

let G be _Graph;
func G .numComponents() -> Cardinal equals :: GLIB_002:def 9
Card (G .componentSet() );
coherence ;

end;

:: deftheorem defines .numComponents() GLIB_002:def 9 :

definition

let G be finite _Graph;
:: original: .numComponents()
redefine func G .numComponents() -> non empty Nat;
coherence

proof end;

end;

definition

let G be _Graph;
let v be Vertex of G;
attr v is cut-vertex means :Def10: :: GLIB_002:def 10
for G2 being removeVertex of G,v holds G .numComponents() <` G2 .numComponents() ;

end;

:: deftheorem Def10 defines cut-vertex GLIB_002:def 10 :

definition

let G be finite _Graph;
let v be Vertex of G;
redefine attr v is cut-vertex means :Def11: :: GLIB_002:def 11
for G2 being removeVertex of G,v holds G .numComponents() < G2 .numComponents() ;
compatibility

proof end;

end;

:: deftheorem Def11 defines cut-vertex GLIB_002:def 11 :

Lm5: for G1 being non trivial connected _Graph
for v being Vertex of G1
for G2 being removeVertex of G1,v st v is endvertex holds
G2 is connected

proof end;

Lm6: for G being _Graph st 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 holds
G is connected

proof end;

Lm7: for G being _Graph st ex v being Vertex of G st G .reachableFrom v = the_Vertices_of G holds
G is connected

proof end;

Lm8: for G being _Graph st G is connected holds
for v being Vertex of G holds G .reachableFrom v = the_Vertices_of G

proof end;

Lm9: for G1, G2 being _Graph
for v1 being Vertex of G1
for v2 being Vertex of G2 st G1 == G2 & v1 = v2 holds
G1 .reachableFrom v1 = G2 .reachableFrom v2

proof end;

Lm10: for G1 being _Graph
for G2 being connected Subgraph of G1 st G2 is spanning holds
G1 is connected

proof end;

Lm11: for G being _Graph holds
( G is connected iff G .componentSet() = {(the_Vertices_of G)} )

proof end;

Lm12: for G1, G2 being _Graph st G1 == G2 holds
G1 .componentSet() = G2 .componentSet()

proof end;

Lm13: for G being _Graph
for x being set st x in G .componentSet() holds
x is non empty Subset of (the_Vertices_of G)

proof end;

Lm14: for G being _Graph
for C being Component of G holds the_Edges_of C = G .edgesBetween (the_Vertices_of C)

proof end;

Lm15: for G being _Graph
for C1, C2 being Component of G holds
( the_Vertices_of C1 = the_Vertices_of C2 iff C1 == C2 )

proof end;

Lm16: for G being _Graph
for C being Component of G
for v being Vertex of G holds
( v in the_Vertices_of C iff the_Vertices_of C = G .reachableFrom v )

proof end;

Lm17: for G being _Graph
for C1, C2 being Component of G
for v being set st v in the_Vertices_of C1 & v in the_Vertices_of C2 holds
C1 == C2

proof end;

Lm18: for G being _Graph holds
( G is connected iff G .numComponents() = 1 )

proof end;

Lm19: for G1, G2 being _Graph st G1 == G2 holds
G1 .numComponents() = G2 .numComponents()
by Lm12;

Lm20: for G being connected _Graph
for v being Vertex of G holds
( not v is cut-vertex iff for G2 being removeVertex of G,v holds G2 .numComponents() <=` G .numComponents() )

proof end;

Lm21: for G being connected _Graph
for v being Vertex of G
for G2 being removeVertex of G,v st not v is cut-vertex holds
G2 is connected

proof end;

Lm22: for G being finite non trivial connected _Graph ex v1, v2 being Vertex of G st
( v1 <> v2 & not v1 is cut-vertex & not v2 is cut-vertex )

proof end;

registration

let G be finite non trivial connected _Graph;
cluster non cut-vertex Element of the_Vertices_of G;
existence

proof end;

end;

Lm23: for G being acyclic _Graph
for W being Path of G
for e being set st not e in W .edges() & e in (W .last() ) .edgesInOut() holds
W .addEdge e is Path-like

proof end;

Lm24: for G being finite non trivial 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 end;

Lm25: for G being finite non trivial Tree-like _Graph ex v1, v2 being Vertex of G st
( v1 <> v2 & v1 is endvertex & v2 is endvertex )

proof end;

registration

let G be finite non trivial Tree-like _Graph;
cluster endvertex Element of the_Vertices_of G;
existence

proof end;

end;

registration

let G be finite non trivial Tree-like _Graph;
let v be endvertex Vertex of G;
cluster -> acyclic Tree-like inducedSubgraph of G,K132((the_Vertices_of G),{v}),G .edgesBetween K132((the_Vertices_of G),{v});
coherence

proof end;

end;

definition

let GSq be GraphSeq;
attr GSq is connected means :Def12: :: GLIB_002:def 12
for n being Nat holds GSq .-> n is connected;
attr GSq is acyclic means :Def13: :: GLIB_002:def 13
for n being Nat holds GSq .-> n is acyclic;
attr GSq is Tree-like means :Def14: :: GLIB_002:def 14
for n being Nat holds GSq .-> n is Tree-like;

end;

:: deftheorem Def12 defines connected GLIB_002:def 12 :

:: deftheorem Def13 defines acyclic GLIB_002:def 13 :

:: deftheorem Def14 defines Tree-like GLIB_002:def 14 :

registration

cluster trivial -> connected ManySortedSet of NAT ;
coherence

proof end;

cluster loopless trivial -> Tree-like ManySortedSet of NAT ;
coherence

proof end;

cluster acyclic -> simple ManySortedSet of NAT ;
coherence

proof end;

cluster Tree-like -> connected acyclic ManySortedSet of NAT ;
coherence

proof end;

cluster connected acyclic -> Tree-like ManySortedSet of NAT ;
coherence

proof end;

end;

registration

cluster halting finite simple connected acyclic Tree-like ManySortedSet of NAT ;
existence

proof end;

end;

registration

let GSq be connected GraphSeq;
let n be Nat;
cluster GSq .-> n -> connected ;
coherence by Def12;

end;

registration

let GSq be acyclic GraphSeq;
let n be Nat;
cluster GSq .-> n -> acyclic ;
coherence by Def13;

end;

registration

let GSq be Tree-like GraphSeq;
let n be Nat;
cluster GSq .-> n -> connected acyclic Tree-like ;
coherence ;

end;

theorem Th2: :: GLIB_002:2

for G being non trivial connected _Graph
for v being Vertex of G holds not v is isolated

proof end;

theorem Th3: :: GLIB_002:3

for G1 being non trivial _Graph
for v being Vertex of G1
for 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 end;

theorem :: GLIB_002:4

for G1 being non trivial connected _Graph
for v being Vertex of G1
for G2 being removeVertex of G1,v st v is endvertex holds
G2 is connected by Lm5;

theorem Th5: :: GLIB_002:5

for G1 being connected _Graph
for W being Walk of G1
for e being set
for G2 being removeEdge of G1,e st W is Cycle-like & e in W .edges() holds
G2 is connected

proof end;

theorem :: GLIB_002:6

for G being _Graph st 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 holds
G is connected by Lm6;

theorem :: GLIB_002:7

for G being trivial _Graph holds G is connected ;

theorem Th8: :: GLIB_002:8

for G1, G2 being _Graph st G1 == G2 & G1 is connected holds
G2 is connected

proof end;

theorem :: GLIB_002:9

for G being _Graph
for v being Vertex of G holds v in G .reachableFrom v by Lm1;

theorem :: GLIB_002:10

for G being _Graph
for x, e, y being set
for v1 being Vertex of G st x in G .reachableFrom v1 & e Joins x,y,G holds
y in G .reachableFrom v1 by Lm2;

theorem :: GLIB_002:11

for G being _Graph
for v being Vertex of G holds G .edgesBetween (G .reachableFrom v) = G .edgesInOut (G .reachableFrom v)

proof end;

theorem :: GLIB_002:12

for G being _Graph
for v1, v2 being Vertex of G st v1 in G .reachableFrom v2 holds
G .reachableFrom v1 = G .reachableFrom v2 by Lm3;

theorem :: GLIB_002:13

for G being _Graph
for v being Vertex of G
for W being Walk of G st v in W .vertices() holds
W .vertices() c= G .reachableFrom v by Lm4;

theorem :: GLIB_002:14

for G1 being _Graph
for G2 being Subgraph of G1
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
G2 .reachableFrom v2 c= G1 .reachableFrom v1

proof end;

theorem :: GLIB_002:15

for G being _Graph st ex v being Vertex of G st G .reachableFrom v = the_Vertices_of G holds
G is connected by Lm7;

theorem :: GLIB_002:16

for G being _Graph st G is connected holds
for v being Vertex of G holds G .reachableFrom v = the_Vertices_of G by Lm8;

theorem :: GLIB_002:17

for G1, G2 being _Graph
for v1 being Vertex of G1
for v2 being Vertex of G2 st G1 == G2 & v1 = v2 holds
G1 .reachableFrom v1 = G2 .reachableFrom v2 by Lm9;

theorem :: GLIB_002:18

for G being _Graph
for v being Vertex of G holds v in G .reachableDFrom v

proof end;

theorem :: GLIB_002:19

for G being _Graph
for x, e, y being set
for v1 being Vertex of G st x in G .reachableDFrom v1 & e DJoins x,y,G holds
y in G .reachableDFrom v1

proof end;

theorem :: GLIB_002:20

for G being _Graph
for v being Vertex of G holds G .reachableDFrom v c= G .reachableFrom v

proof end;

theorem :: GLIB_002:21

for G1 being _Graph
for G2 being Subgraph of G1
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
G2 .reachableDFrom v2 c= G1 .reachableDFrom v1

proof end;

theorem :: GLIB_002:22

for G1, G2 being _Graph
for v1 being Vertex of G1
for v2 being Vertex of G2 st G1 == G2 & v1 = v2 holds
G1 .reachableDFrom v1 = G2 .reachableDFrom v2

proof end;

theorem :: GLIB_002:23

for G1 being _Graph
for G2 being connected Subgraph of G1 st G2 is spanning holds
G1 is connected by Lm10;

theorem :: GLIB_002:24

for G being _Graph holds union (G .componentSet() ) = the_Vertices_of G

proof end;

theorem :: GLIB_002:25

for G being _Graph holds
( G is connected iff G .componentSet() = {(the_Vertices_of G)} ) by Lm11;

theorem :: GLIB_002:26

for G1, G2 being _Graph st G1 == G2 holds
G1 .componentSet() = G2 .componentSet() by Lm12;

theorem :: GLIB_002:27

for G being _Graph
for x being set st x in G .componentSet() holds
x is non empty Subset of (the_Vertices_of G) by Lm13;

theorem :: GLIB_002:28

for G being _Graph holds
( G is connected iff G .numComponents() = 1 ) by Lm18;

theorem :: GLIB_002:29

for G1, G2 being _Graph st G1 == G2 holds
G1 .numComponents() = G2 .numComponents() by Lm19;

theorem :: GLIB_002:30

for G being _Graph holds
( G is Component of G iff G is connected )

proof end;

theorem :: GLIB_002:31

for G being _Graph
for C being Component of G holds the_Edges_of C = G .edgesBetween (the_Vertices_of C) by Lm14;

theorem :: GLIB_002:32

for G being _Graph
for C1, C2 being Component of G holds
( the_Vertices_of C1 = the_Vertices_of C2 iff C1 == C2 ) by Lm15;

theorem :: GLIB_002:33

for G being _Graph
for C being Component of G
for v being Vertex of G holds
( v in the_Vertices_of C iff the_Vertices_of C = G .reachableFrom v ) by Lm16;

theorem :: GLIB_002:34

for G being _Graph
for C1, C2 being Component of G
for v being set st v in the_Vertices_of C1 & v in the_Vertices_of C2 holds
C1 == C2 by Lm17;

theorem :: GLIB_002:35

for G being connected _Graph
for v being Vertex of G holds
( not v is cut-vertex iff for G2 being removeVertex of G,v holds G2 .numComponents() <=` G .numComponents() ) by Lm20;

theorem :: GLIB_002:36

for G being connected _Graph
for v being Vertex of G
for G2 being removeVertex of G,v st not v is cut-vertex holds
G2 is connected by Lm21;

theorem :: GLIB_002:37

for G being finite non trivial connected _Graph ex v1, v2 being Vertex of G st
( v1 <> v2 & not v1 is cut-vertex & not v2 is cut-vertex ) by Lm22;

theorem Th38: :: GLIB_002:38

for G being _Graph
for v being Vertex of G st v is cut-vertex holds
not G is trivial

proof end;

theorem :: GLIB_002:39

for G1, G2 being _Graph
for v1 being Vertex of G1
for v2 being Vertex of G2 st G1 == G2 & v1 = v2 & v1 is cut-vertex holds
v2 is cut-vertex

proof end;

theorem Th40: :: GLIB_002:40

for G being finite connected _Graph holds G .order() <= (G .size() ) + 1

proof end;

theorem :: GLIB_002:41

for G being acyclic _Graph holds G is simple ;

theorem :: GLIB_002:42

for G being acyclic _Graph
for W being Path of G
for e being set st not e in W .edges() & e in (W .last() ) .edgesInOut() holds
W .addEdge e is Path-like by Lm23;

theorem :: GLIB_002:43

for G being finite non trivial 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 Lm24;

theorem Th44: :: GLIB_002:44

for G1, G2 being _Graph st G1 == G2 & G1 is acyclic holds
G2 is acyclic

proof end;

theorem :: GLIB_002:45

for G being finite non trivial Tree-like _Graph ex v1, v2 being Vertex of G st
( v1 <> v2 & v1 is endvertex & v2 is endvertex ) by Lm25;

theorem Th46: :: GLIB_002:46

for G being finite _Graph holds
( G is Tree-like iff ( G is acyclic & G .order() = (G .size() ) + 1 ) )

proof end;

theorem :: GLIB_002:47

for G being finite _Graph holds
( G is Tree-like iff ( G is connected & G .order() = (G .size() ) + 1 ) )

proof end;

theorem Th48: :: GLIB_002:48

for G1, G2 being _Graph st G1 == G2 & G1 is Tree-like holds
G2 is Tree-like

proof end;

theorem :: GLIB_002:49

for G being _Graph
for x being set st G is_DTree_rooted_at x holds
x is Vertex of G

proof end;

theorem :: GLIB_002:50

for G1, G2 being _Graph
for x being set st G1 == G2 & G1 is_DTree_rooted_at x holds
G2 is_DTree_rooted_at x

proof end;