On the homotopy type of complexes of graphs with bounded domination number

2019/01/22 by González, Jesús, Hoekstra-Mendoza, Teresa I.
#05C69 #55P15 #55U10 #57M15 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.07130

Abstract

Let D_n,γ be the complex of graphs on n vertices and domination number at least γ. We prove that D_n,n-2 has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of the needed matching is proved by introducing a relativized form of a well known method for constructing acyclic matchings on suitable chunks of simplices. Our approach allows us to extend our results to the realm of infinite graphs. In addition, we give evidence supporting the assertion that the homotopy equivalences D_n,n-1≃\bigvee S⁰ and D_n,n-2≃\bigvee S² do not seem to generalize for D_n,γ with γ≤ n-3.

Related

• Discrete Morse theory and the topology of matching complexes of complete graphs
• 2-Matching Complexes
• Complexes of graphs with bounded matching size
• Matching complexes of \bf 3 × n grid graphs
• Higher matching complexes of complete graphs and complete bipartite graphs