2022/12/02 by Tung Nguyen, Nguyen, Tung, Alex Scott +3 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Limits and Structures in Graph Theory #Complexity and Algorithms in Graphs
paper · pdf · doi:10.48550/arxiv.2212.01089
Let us say a graph is sO-free, where s≥ 1 is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is 2O-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that 2O-free graphs have only a polynomial number of induced paths. In this paper we prove Le's conjecture; indeed, we will show that for all s≥ 1, there exists c>0 such that every sO-free graph G has at most |G|c induced paths. This provides a poly-time algorithm to test if a graph is sO-free, for all fixed s. The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every sO-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in |G|. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.