:: GLIB_002 semantic presentation
theorem Th1: :: GLIB_002:1
:: deftheorem Def1 defines connected GLIB_002:def 1 :
:: deftheorem Def2 defines acyclic GLIB_002:def 2 :
:: deftheorem Def3 defines Tree-like GLIB_002:def 3 :
:: deftheorem Def4 defines is_DTree_rooted_at GLIB_002:def 4 :
:: deftheorem Def5 defines .reachableFrom GLIB_002:def 5 :
:: deftheorem Def6 defines .reachableDFrom GLIB_002:def 6 :
Lemma91:
for G being _Graph
for v being Vertex of G holds v in G .reachableFrom v
Lemma92:
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
Lemma95:
for G being _Graph
for v1, v2 being Vertex of G st v1 in G .reachableFrom v2 holds
G .reachableFrom v1 = G .reachableFrom v2
Lemma98:
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
:: deftheorem Def7 defines Component-like GLIB_002:def 7 :
:: deftheorem Def8 defines .componentSet() GLIB_002:def 8 :
:: deftheorem Def9 defines .numComponents() GLIB_002:def 9 :
:: deftheorem Def10 defines cut-vertex GLIB_002:def 10 :
:: deftheorem Def11 defines cut-vertex GLIB_002:def 11 :
Lemma109:
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
Lemma124:
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
Lemma125:
for G being _Graph st ex v being Vertex of G st G .reachableFrom v = the_Vertices_of G holds
G is connected
Lemma126:
for G being _Graph st G is connected holds
for v being Vertex of G holds G .reachableFrom v = the_Vertices_of G
Lemma127:
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
Lemma128:
for G1 being _Graph
for G2 being connected Subgraph of G1 st G2 is spanning holds
G1 is connected
Lemma131:
for G being _Graph holds
( G is connected iff G .componentSet() = {(the_Vertices_of G)} )
Lemma132:
for G1, G2 being _Graph st G1 == G2 holds
G1 .componentSet() = G2 .componentSet()
Lemma133:
for G being _Graph
for x being set st x in G .componentSet() holds
x is non empty Subset of (the_Vertices_of G)
Lemma134:
for G being _Graph
for C being Component of G holds the_Edges_of C = G .edgesBetween (the_Vertices_of C)
Lemma139:
for G being _Graph
for C1, C2 being Component of G holds
( the_Vertices_of C1 = the_Vertices_of C2 iff C1 == C2 )
Lemma142:
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 )
Lemma146:
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
Lemma147:
for G being _Graph holds
( G is connected iff G .numComponents() = 1 )
Lemma149:
for G1, G2 being _Graph st G1 == G2 holds
G1 .numComponents() = G2 .numComponents()
by ;
Lemma150:
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() )
Lemma151:
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
Lemma152:
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 )
Lemma274:
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
Lemma277:
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 )
Lemma281:
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 )
:: 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 :
theorem Th2: :: GLIB_002:2
theorem Th3: :: GLIB_002:3
theorem Th4: :: GLIB_002:4
theorem Th5: :: GLIB_002:5
theorem Th6: :: GLIB_002:6
theorem Th7: :: GLIB_002:7
theorem Th8: :: GLIB_002:8
theorem Th9: :: GLIB_002:9
theorem Th10: :: GLIB_002:10
theorem Th11: :: GLIB_002:11
theorem Th12: :: GLIB_002:12
theorem Th13: :: GLIB_002:13
theorem Th14: :: GLIB_002:14
theorem Th15: :: GLIB_002:15
theorem Th16: :: GLIB_002:16
theorem Th17: :: GLIB_002:17
theorem Th18: :: GLIB_002:18
theorem Th19: :: GLIB_002:19
theorem Th20: :: GLIB_002:20
theorem Th21: :: GLIB_002:21
theorem Th22: :: GLIB_002:22
theorem Th23: :: GLIB_002:23
theorem Th24: :: GLIB_002:24
theorem Th25: :: GLIB_002:25
theorem Th26: :: GLIB_002:26
theorem Th27: :: GLIB_002:27
theorem Th28: :: GLIB_002:28
theorem Th29: :: GLIB_002:29
theorem Th30: :: GLIB_002:30
theorem Th31: :: GLIB_002:31
theorem Th32: :: GLIB_002:32
theorem Th33: :: GLIB_002:33
theorem Th34: :: GLIB_002:34
theorem Th35: :: GLIB_002:35
theorem Th36: :: GLIB_002:36
theorem Th37: :: GLIB_002:37
theorem Th38: :: GLIB_002:38
theorem Th39: :: GLIB_002:39
theorem Th40: :: GLIB_002:40
theorem Th41: :: GLIB_002:41
theorem Th42: :: GLIB_002:42
theorem Th43: :: GLIB_002:43
theorem Th44: :: GLIB_002:44
theorem Th45: :: GLIB_002:45
theorem Th46: :: GLIB_002:46
theorem Th47: :: GLIB_002:47
theorem Th48: :: GLIB_002:48
theorem Th49: :: GLIB_002:49
theorem Th50: :: GLIB_002:50