The Perfect Matching Reconfiguration Problem

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time

TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs

We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S= of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.Comment: This work was supported by ANR project GrR (ANR-18-CE40-0032) and submitted to the conference WADS 202

Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar