2021/09/09 by Okura, Kengo
#05C69 #05C76 #55P15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2109.04181
We study the independence complex of the lexicographic product G[H] of a forest G and a graph H. We prove that for a forest G which is not dominated by a single vertex, if the independence complex of H is homotopy equivalent to a wedge sum of spheres, then so is the independence complex of G[H]. We offer two examples of explicit calculations. As the first example, we determine the homotopy type of the independence complex of Lm [H], where Lm is the tree on m vertices with no branches, for any positive integer m when the independence complex of H is homotopy equivalent to a wedge sum of n copies of d-dimensional sphere. As the second one, for a forest G and a complete graph K, we describe the homological connectivity of the independence complex of G[K] by the independent domination number of G.