Posets, clique graphs and their homotopy type

To any finite poset P we associate two graphs which we denote by Ω(P) and ℧(P). Several standard constructions can be seen as Ω(P) or ℧(P) for suitable posets P, including the comparability graph of a poset, the clique graph of a graph and the 1-skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we study and compare the homotopy types of Ω(P), ℧(P) and P. As our main application we obtain a theorem, stronger than those previously known, giving sufficient conditions for a graph to be homotopy equivalent to its clique graph. We also introduce a new graph operator H that preserves clique-Hellyness and dismantlability and is such that H(G) is homotopy equivalent to both its clique graph and the graph G.