AbstractLet L be a subcomplex of a complex K. If the homomorphism from inclusion i∗:Hq(L)→Hq(K) is an isomorphism for all q ⩾ 0, then we say that L and K are endo-homologous. The clique complex of a graph G, denoted by C(G), is an abstract complex whose simplices are the cliques of G. The present paper is a generalization of Ivashchenko (1994) along several directions. For a graph G and a given subgraph F of G, some necessary and sufficient conditions for C(G) to be endo-homologous to C(F) are given. Similar theorems hold also for the independence complex I(G) of G, where I(G) − C(Gc), the clique complex of the complement of G

Liu, Guizhen

Xie, Litong

Xu, Baogang

Elsevier - Publisher Connector 

ELSEVIER Discrete Mathematics 188 (1998) 285-291 
DISCRETE 
MATHEMATICS 
Note 
On endo-homology of complexes of graphs 1 
Litong Xie, Guizhen Liu, Baogang Xu* 
Institute of Mathematics. Shandong University, Jinan. 250100, China 
Received 26 May 1997; revised 20 October 1997; accepted 20 January 1998 
Abstract 
Let L be a subcomplex of a complex K. If the homomorphism from inclusion i.:Hq(L)H 
Hq(K) is an isomorphism for all q>~0, then we say that L and K are endo-homologous. The clique 
complex of a graph G, denoted by C(G), is an abstract complex whose simplices are the cliques 
of G. The present paper is a generalization f lvashchenko (1994) along several directions. For a 
graph G and a given subgraph F of G, some necessary and sufficient conditions for C(G) to be 
endo-homologous to C(F) are given. Similar theorems hold also for the independence omplex 
I(G) of G, where I(G)= C(GC), the clique complex of the complement of G. (~) 1998 Elsevier 
Science B.V. All rights reserved 
Ke)'word~: Clique; Finite complex; Homology; Endo-homology 
1. Introduction 
All graphs considered are finite and simple, all complexes considered are finite and 
simplicial. Undefined terms can be found in [1,2]. 
Given a n-complex K (n-complex means K is a n-dimensional complex) and a sub- 
complex L of  K. For each q (O<~q<~n), we use Hq(K,A) and Hq(K,L;A) to denote 
the qth homology group of K and the qth relative homology group of K modulo L, re- 
spectively, where A is the coefficient group, which may be the group of  integers or any 
other abelian group. Without confussion, we briefly denote Hq(K,A) and Hq(K,L;A) 
by Hq(K) and Hq(K,L), respectively. 
The augmented complex [4] of  complex K, denoted by K +, is a complex obtained 
from K by adding a -1-dimensional  simplex to K, which is the face of  every simplex 
t Research supported by the Doctorial Fundation of the Education Committee of P.R. China. 
* Corresponding author. E-mail: math@sdu.edu.cn. 
0012-365X/98/$19.00 Copyright (~) 1998 Elsevier Science B.V. All rights reserved 
PH SOO12-365X( 98 )00028-4 
286 L. Xie et al./Discrete Mathematics 188 (1998) 285-291 
of K and we have 
H_ I (K+)=0,  Ho(K)~A + Ho(K+), Hq(K+)=Hq(K) (q > O) 
when K is not empty and 
H_I(K+)-~A, Hq(K+)=Hq(K) (q>~O) 
when K is empty [4]. 
A complex K is said to be acyclic, if Hq(K+)=O for each q~> - 1. Obviously, K 
is acyclic, if and only if Hq(K)=0 (q>~ 1) and Ho(K)~--A. Thus, an acyclic complex 
must be nonempty and connected. 
Given a graph G=(V(G),E(G)), where V(G) and E(G) denote the vertex-set and 
edge-set of G, respectively. G[S] denotes the subgraph of G induced by S C V(G). 
No(u) denotes the set of vertices adjacent o u in G. Without confusion, we still 
use N6(u) to denote the subgraph of G induced by No(u). S C V(G) is called an 
independent set of G if E(G[S]) = ~. A subgraph H of G is called a clique of G if H 
is a complete graph. We use G c to denote the complement graph of G. 
There are various abstract complexes [2] related to a simple graph [1] G. Notable 
are the following: 
1. The neibourhood complex [5] of G, each of whose simplices is formed by the 
neighbour set of a vertex of G. 
2. The independence complex of G, denoted by I(G), whose simplices are the inde- 
pendent sets of G. 
3. The clique complex of G, denoted by C(G), whose simplices are the cliques 
of G. 
We are interested in the last two kinds by obvious reason that isomorphic graphs have 
isomorphic lique complexes and vice versa [8]. Since C(G)=I(G c) and I(G)=C(GC), 
we need only to study the clique complexes of graphs. The results on independence 
complexes can be obtained analogously. 
The present paper may be considered as a generalization of [3] along several direc- 
tions. In [3], the homology groups of a graph G are defined, which, in fact, are the 
homology groups of the clique complex C(G) of G. For convenience, we do not make 
distinction between an abstract complex and its geometrical realization as a simplicial 
complex K in EucBdean spaces, so we may apply the usual theorems about simplicial 
complex [2,4] to I(G) and C(G) freely. 
Let L be a subcomplex of a complex K. If  the homomorphism from inclusion 
i, : Hq(L ) ~-~ Hq(K) 
is an isomorphism for all q, then we say that L and K are endo-homologous to 
each other. In general, if the homology groups of K and L are isomorphic, i.e., 
Hq(L)~-Hq(K) (q>~0), then we say that L and K are homologous. Obviously, if L 
and K are endo-homologous, they are surely homologous; but the converse may not 
be true. For example, let G be the graph shown in the following figure and cycle 
L. Xie et al./Discrete Mathematics 188 (1998) 285-291 287 
F=VlV2V3V4V5Vl be a subgraph of G. Then C(F) and C(G) are homologous but not 
endo-homologous. In fact, i. maps the 1-cycle of C(F) to zero. 
I 7)3 
2. Main theorems 
Theorem 1. Let F be an induced subgraph of a given graph G. Graph G & obtained 
from G by adding a new vertex v and joining it to each vertex of V(F). Then, C(F) 
and C(G) are endo-homologous, if and only if C(G) is aeyclic. 
Theorem 1'. Let F be an induced subgraph of a given graph G. Graph 0 is obtained 
from G by adding a new vertex v and joining it to each vertex of V(G)\V(F). Then, 
I(F) and I(G) are endo-homologous, if and only if I(G) is acylic. 
Theorem 2. Let G be a graph and v be an arbitrary vertex of G. The clique com- 
plexes C(G-  v) and C(G) are endo-homologous, if and only if C(NG(V)) is acyclic, 
where NG(V) denotes the subgraph of G induced by the neighbour set of v 
inG. 
Theorem 2'. Let G be a graph, v be an arbitrary vertex of G and L'(v) = G - v - 
NG(V). The independence omplexes I(G - v) and I(G) are endo-homologous, if and 
only if I(L'(v)) is acyclic. 
Theorem 3. Let uv be an edge of a given graph G. The clique complexes C(G-  uv) 
and C(G) are endo-homologous, if and only if C(NG(v)NNG(U)) is acyclic, where 
NG(V) N NG(U) denotes the subgraph of G induced by the common eighbour set of v 
and u in G. 
Theorem 3'. Let u and v be two non-adjacent vertices of a given graph G. The 
independence complexes I(G + uv) and I(G) are endo-homologous, if and only if 
I(L'G(V ) ALe(u)) is acyclic. 
288 L. Xie et al./Discrete Mathematics 188 (1998) 285-291 
3. Some useful iemmas 
To prove our main theorems, we need the following lemmas. 
Lemma 1 [4,6,7]. Let L be a subcomplex of complex K. Then L and K are endo- 
homologous if and only if 
Hq(K,L)~O, q>~O. 
Proof. This is a direct consequence of the exactness of the homology sequence of 
K and L. [] 
Lennna 2 [2,4]. Let L be a subcomplex of a complex K,v be a single O-simplex 
which is not in K. v o L represents the cone on L with vertex v, Iq = K U v o L. Then 
Hq(K+)~Hq(K,L) q>/O. 
Lemma 3 [2,4]. Let L be a complex and v be a single O-simplex which is not in L. 
Then 
Hq(voL, L)~Hq_I(L +) q>~O. 
Lemma 4. Let L be a complex, u and v be distinct O-simplices which are not in L. 
Then 
Hq(uoL, L)~Hq((uoLUvoL)+),  q>~O. 
Proof. Let K = u o L, /~ = K U v o L. By Lemma 2, for q >t 0, 
Hq( (u o L U v o L )+ I= Hq(I~+ )~-- Hq(K, L I = Hq(u o L, L ). 
Lemma 5. Let K be a cone of dimension and L be a nonempty subcomplex of K. 
Then 
~. :Hq(K,L)~--+Hq_I(L+), q>~O 
are all isomorphisms. 
Proof. We know that the following homology sequence is exact provided L is non- 
empty. 
O~--~Hn(L) ~z~ Hn(K) ~z~ Hn(K,L) ~ Hn-1(L) i. Hn-I(K) 
~ ... f-~ Hq(L) ~ Hq(K) ~ Hq(K,L) ~ Hq-I(L) ~ "-  
~z~ H~(K,L) ~ Ho(L +) ~ Ho(K + ) J~ Ho(K,L) ~ O, ( I )  
L. Xie et al./Discrete Mathematics 188 (1998) 285 291 289 
where iq* are homomorphisms from inclusion; jq,, homomorphisms from j; 0q*, homo- 
morphisms from a; and i, + and j,+ are reduced homomorphisms. 
Since K is a cone, we have for each q>~O, Hq(K+)~-O. Since L is nonempty, we 
have H_ 1 (L +) = 0. 
From these facts and the exactness of the homology sequence (1), it is easy to see 
that our lemma holds. [] 
4. Proofs of main theorems 
Proof of Theorem 1. Let K = C(G), L = C(F) and /< -- C(G). Because G = G + v, 
where N~(v)= V(F), then 
I< = C(G) U v o C(F) = K U v o L. (2) 
By Lemma 2, 
He(I<+)-~Hq(K,L), q>~O, 
i.e., 
Hq(C(UJ)+)~Hq(C(G),C(F)), q>~O. (3) 
By Lemma 1 and (3), C(F) and C(G) are endo-homologous if and only if 
O'~Hq(C(GI, C(F))~-Hq(C(G+)), q>~O. 
The theorem immediately follows because that C(G) is a nonempty complex. [] 
Proof of Theorem 2. By Lemma 1, C(G - v) and C(G) are endo-homologous if and 
only if 
Hq(C(G) ,C(G-  v))~-O, q>~O. (4) 
Let No(v)=Nc(v)+v. Then, C(Nc(v))= v o C(Nc(v)), C(G)= C(G-v)  U C(No(v)). 
In other words, we have 
C(G) = C(G - v) U v o C(NG(v)) 
and 
C(G - v) n v o C(Nd(v) )  = C(No(v ) ) .  
According to Excision-Theorem and Lemma 3, 
Hq( C( G), C( G - v) ) = Hq(v o C(NG(V) ), C(NG(v) ) 
-~Hq_l(C(NG(v))+), q>~O. (5) 
290 L Xie et al./Discrete Mathematics 188 (1998) 285-291 
Combining (4) and (5), C(G - v) and C(G) are endo-homologous if and only if 
Hq(C(NG(v))+)~-O, q>~ - 1, 
i.e., if and only if C(Nc(v) )  is an acyclic complex. [] 
Proof of Theorem 3. Let F = C(Nc(u)ANt (v ) ) .  It is obvious that 
C(G)  = C(G - uv) U C(Nc(u)  N No(v)), 
C(G - uv) O C(Nc(u)  n No(v))  = u o F U v o F 
and 
C(Na(u) n NG(v)) = u o (v oF)  = v o (u oF). 
According to Excision-Theorem, Lemma 5 and (6)-(8), 
(6) 
(7) 
(8) 
Hq(C(G), C(G - uv)) ~ Hq(u o v o F, u o F U v o F )  
"~Hq_ l ( (uoFUvoF)+) ,  q>~O (9) 
By Lemmas 3 and 4, 
Hq((uofUvof )+)~- -Hq_ l (F+) ,  q>~O. (10) 
Combining (9) and (10), we have 
Hq(C(G) ,C(G-  uv))~-Hq_2(F+), q>~l. (11) 
By Lemma 1, C(G - uv) and C(G) are endo-homologous if and only if 
Hq(C(G) ,C(G-uv) ) -~O,  q )O.  (12) 
Noticing that, when F is nonempty, uoFUvoF  is connected and Ho(C(G), 
C (G-  uv))~-O by (9), our theorem immediately follows from (11) and (12). [] 
Note: The clique complexes of contractible graphs defined in [3] are all proved to 
be acyclic. Among the four kinds of contractible transformations in [3] which do not 
change the homology groups of graphs, the first two transformations, i.e., deleting or 
gluing of a vertex v, are to transform G to G-  v or, conversely, transform G-  v to G, 
when the induced subgraph Na(v) in G is contractible. The other two transformations 
are applied to G and G-  uv, when N6(u)NNc(v)  is a contractible graph. Thus, the 
main theorems in [3] are special cases of the sufficient parts of Theorems 2 and 3 
above. 
Finally, we propose the following. 
Conjecture. The clique complexes of graphs G and H are homologous, if and only if 
there exists a finite number of graphs, 
G = Fo,F1 . . . . .  Fn = H (n>~ 1), 
L. Xie et al./Discrete Mathematics 188 (1998) 285-291 291 
such that, for each i (O<~i<<.n- 1), either Fi is isomorphic to Fi+l, or one of the two 
graphs Fi and Fi+l is a subgraph of the other with C(Fi) being endo-homologous to 
C(Fi+I ). Similar conjecture can also be made for the independence omplexes of G 
and H. 
References 
[1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press, London, 1976. 
[2] P.J. Giblin, Graphs, Surfaces and Homology, Chapman & Hall, London, 1981. 
[3] A.V. Ivashchenko, Contractible transformations do not change the homology groups of graphs, Discrete 
Math. 126 (1994) 159-170. 
[4] C.H. Jiang, Introduction to Topology, Shanghai Publishing Company of Science and Technology, 1978 
(in Chinese). 
[5] L. Lovasz, Kneser's conjecture, Chromatic number and homotopy, J. Combin. Theory Ser. A 25 (1978) 
319-324. 
[6] Y. Peng, On the invariance of the homology groups of the neighbourhood complex of a simple graph, 
Ann. Math. I lA (6) (1990) 677-682 (in Chinese). 
[7] L. Xie, J. Liu, G. Liu, Graphs and Algebraic Topology, Shandong University Press, 1994 (in Chinese). 
[8] L. Xie, On independence omplexes of graphs. Graphs and Combinatorics'95, vol. 2, World Scientific 
Press, Singapore, to appear. 


On endo-homology of complexes of graphs 

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