Efficient Enumeration of Induced Matchings in a Graph without Cycles with Length Four

Abstract

In this study, we address a problem pertaining to the induced matching enumeration. An edge set M is an induced matching of a graph G=(V,E). The enumeration of matchings has been widely studied in literature; however, there few studies on induced matching. A straightforward algorithm takes O(Δ²) time for each solution that is coming from the time to generate a subproblem, where Δ is the maximum degree in an input graph. To generate a subproblem, an algorithm picks up an edge e and generates two graphs, the one is obtained by removing e from G, the other is obtained by removing e, adjacent edge to e, and edges adjacent to adjacent edge of e. Since this operation needs O(Δ²) time, a straightforward algorithm enumerates all induced matchings in O(Δ²) time per solution. We investigated local structures that enable us to generate subproblems within a short time and proved that the time complexity will be O(1) if the input graph is C₄-free. A graph is C₄-free if and only if none of its subgraphs have a cycle of length four.