On endo-homology of complexes of graphs.

*(English)*Zbl 0955.05104Summary: 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) |

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\textit{L. Xie} et al., Discrete Math. 188, No. 1--3, 285--291 (1998; Zbl 0955.05104)

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