Homotopyc properties of the clique operator in graphs
Abstract
This work shows the results obtained in my undergraduate thesis, where combinaorial topology concepts are addressed. Mainly we work with the simplicial complex of completes of a graph, which in turn, we can associate it a topological space through its geometric realization. In this way, topological concepts are associated with graphs that are combinatorial structures, thus, we are interested in the type of homotopy that the graphs have. The clique operator K associates to each graph G another graph that is denoted by K(G) and which is known like the clique graph of G. When G is homotopic to K(G) it says that G is homotopically invariant. There are “large” classes of graphs that already known to be homotopically invariant, such as those classes of graphs that satisfy the clique Helly property or the classes of dismantlable graphs. This article will analyze some conditions that guarantee that the graphs G that satisfy that K(G) is clique Helly are homotopically invariant, all this making use of elementary collapses.
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References
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