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On endo-homology of complexes of graphs. (English) Zbl 0955.05104
Summary: Let $$L$$ be a subcomplex of a complex $$K$$. If the homomorphism from inclusion $$i_*: H_q(L)\mapsto H_q(K)$$ is an isomorphism for all $$q\geq 0$$, 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 A. V. Ivashchenko [Discrete Math. 126, No. 1-3, 159-170 (1994; Zbl 0798.05027)] 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 $$I(G)= C(G^{\text{c}})$$, the clique complex of the complement of $$G$$.
##### MSC:
 05C99 Graph theory 05E25 Group actions on posets, etc. (MSC2000)
##### Keywords:
endo-homology; complex
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##### References:
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