Higher independence complexes of graphs

Abstract: In 2006, Szabó and Tardos generalized the concept of independence complex by defining $r$-independence complex of a graph $G$ for any $r \geq 1$. Independence complexes have applications in several areas. The topology of independence complex is related to many combinatorial properties of the underlined graph. The $r$-independence complex of $G$, denoted Ind$_r(G)$, is the simplicial complex whose simplices are those subsets $I \subseteq V(G)$ such that each connected component of the induced subgraph $G[I]$ has at most $r$ vertices.

In this talk, we give a lower bound for the distance $r$-domination number of the graph $G$ (which is a very well studied notion in graph theory and a natural generalization of the domination number of the graph) in terms of the homological connectivity of the Ind$_r(G)$. We also prove that Ind$_r(G)$, for a chordal graph $G$, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose $r$-independence complex has the homotopy type of the given wedge. This is a joint work with Anurag Singh and Priyavrat Deshpande.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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