On endo-homology of complexes of graphs

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