Independent Feedback Vertex Set for $$P_5$$ P 5 -Free Graphs

Abstract

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k ≥ 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 k ≥ 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. Keywords Independent feedback vertex set • Near-bipartiteness • Independent odd cycle transversal • 3-Colouring This paper received support from EPSRC (EP/K025090/1), London Mathematical Society (41536), the Leverhulme Trust (RPG-2016-258) and Fondation Sciences Mathématiques de Paris.

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

“…We note finally that that there are other similar transversal problems that have been studied, but their complexity classifications on H-free graphs have not been settled: (Subset) Even Cycle Transversal [18,21,24], for example. Versions of the transversal problems that we have considered that have the additional constraint that the transversal must induce either a connected graph or an independent set have also been studied for H-free graphs [4,8,13,17]. An interesting direction for further research is to consider the subset variant of these problems, and, more generally, to understand the relationships amongst the computational complexities of all these problems.…”

Faster 3-Coloring of Small-Diameter Graphs

“…We observe that our approach can be used to obtain subexponential-time algorithms for the problem of finding a list homomorphism from a graph with diameter at most 3 to certain graphs, including in particular all cycles. We refer to Section 5.1 for the definition of the problem and the precise statement of our results; let us just point out that under the ETH the problems considered there cannot be solved in subexponential time in general graphs [13,14] We conclude with discussing the possibility of extending our algorithms to weighted coloring problems, with Independent Odd Cycle Transversal [4] as a prominent special case.…”