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

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.