On The Complexity of Matching Cut for Graphs of Bounded Radius and H-Free Graphs
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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 P_6-free graphs, extending a recent result of Feghali for P_5-free graphs. We then extend our result to hold even for (sP_3+P_6)-free graphs for every s≥ 0 and initiate a complexity classification of Matching Cut for H-free graphs.
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