Independent Feedback Vertex Set for P_5-free Graphs
The NP-complete problem Feedback Vertex Set is to decide if 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 if 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 in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.
feedback vertex set
odd cycle transversal
independent set
H-free graph
16:1-16:12
Regular Paper
Marthe
Bonamy
Marthe Bonamy
Konrad K.
Dabrowski
Konrad K. Dabrowski
Carl
Feghali
Carl Feghali
Matthew
Johnson
Matthew Johnson
Daniël
Paulusma
Daniël Paulusma
10.4230/LIPIcs.ISAAC.2017.16
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