Posets, clique graphs and their homotopy type

To any finite poset P we associate two graphs which we denote by View the MathML source and ?(P). Several standard constructions can be seen as View the MathML source 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 View the MathML source, ?(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.