A note on Matching-Cut in $P_t$-free Graphs

Abstract

A matching-cut of a graph is an edge cut that is a matching. The problem matching-cut is that of recognizing graphs with a matching-cut and is NP-complete, even if the graph belongs to one of a number of classes. We initiate the study of matching-cut for graphs without a fixed path as an induced subgraph. We show that matching-cut is in P for P 5 -free graphs, but that there exists an integer t > 0 for which it is NP-complete for P t -free graphs.