Abstract
The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer kge 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains polynomial-time solvable for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer kge 0. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.
Highlights
Many computational problems in the theory and application of graphs can be formulated as modification problems: from a graph G, some other graph H with a desired property must be obtained using certain permitted operations
Our main result is that Independent Feedback Vertex Set is polynomialtime solvable for P5-free graphs
We prove that our results for Independent Feedback Vertex Set hold for Independent Odd Cycle Transversal
Summary
Many computational problems in the theory and application of graphs can be formulated as modification problems: from a graph G, some other graph H with a desired property must be obtained using certain permitted operations. The computational complexity of a graph modification problem depends on the desired property, the operations allowed and the possible inputs; that is, we can prescribe the class of graphs to which G must belong. This leads to a rich variety of different problems, which makes graph modification a central area of research in algorithmic graph theory. The Feedback Vertex Set problem asks whether or not a graph has a feedback vertex set of size at most k for some integer k ≥ 0 and is a well-known example of a graph modification problem: the desired property is that the obtained graph is acyclic and the permitted operation is vertex deletion. We survey known results on Independent Feedback Vertex Set below
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