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تسجيل مستخدم جديدFeedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs
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We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for $sP_3$-free graphs for every integer $sgeq 1$. Our results show that both problems exhibit the same behaviour on $H$-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph $G$ a largest induced subgraph whose blocks belong to some finite class ${cal C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.
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A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an
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