Computing Weighted Subset Transversals in H-Free Graphs
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For the Odd Cycle Transversal problem, the task is to find a small set S of vertices in a graph that intersects every cycle of odd length. The generalization Subset Odd Cycle Transversal requires that S only intersects those odd cycles that include a vertex of a distinguished subset T of the vertex set. If we are also given weights for the vertices of the graph, we can ask instead that S has small weight: this is the problem Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity dichotomy for this problem when the input is restricted to graphs that do not contain a graph H as an induced subgraph. In particular, we show that for (3P_1+P_2)-free graphs (where P_r is the path on r vertices) there is a polynomial-time algorithm, but the problem is NP-complete for 5P_1-free graphs, that is, graphs of independence number 4. Thus we obtain a dichotomy with respect to the independence number; this is an analogue of the dichotomy for Weighted Subset Feedback Vertex Set recently obtained by Papadopoulos and Tzimas. In contrast, Subset Feedback Vertex Set and Subset Odd Cycle Transversal have a polynomial-time algorithm for any graph class with bounded independence number. We also generalize the polynomial-time result of Papadopoulos and Tzimas for Weighted Subset Feedback Vertex Set on 4P_1-free graphs to (3P_1+P_2)-free graphs. As a consequence, we show that the complexity for both of the weighted subset transversal problems restricted to H-free graphs remains open for just three particular graphs H.
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