On the complexity of matching cut for graphs of bounded radius and H-free graphs

Abstract

For a connected graph G=(V,E), a matching M⊆E is a matching cut of G if G−M is disconnected. It is known that for an integer d, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most d if d≤2 and NP-complete if d≥3. We prove the same dichotomy for graphs of bounded radius. For a graph H, a graph is H-free if it does not contain H as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for P6-free graphs, extending a recent result of Feghali for P5-free graphs. We then extend our result to hold even for (sP3+P6)-free graphs for every s≥0 and initiate a complexity classification of Matching Cut for H-free graphs.