Abstract
The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.
Highlights
A graph is near-bipartite if its vertex set can be partitioned into sets A and B, where A is an independent set and B induces a forest
In [3] we proved that finding a minimum independent feedback vertex set is polynomial-time solvable even for P5-free graphs
We prove that Independent Feedback Vertex Set is polynomial-time solvable for graphs of diameter 2
Summary
A graph is near-bipartite if its vertex set can be partitioned into sets A and B, where A is an independent set and B induces a forest. We prove that Independent Feedback Vertex Set is polynomial-time solvable for graphs of diameter 2 This generalises the result of Yang and Yuan [16] for Near-Bipartiteness restricted to graphs of diameter 2. Yang and Yuan [16] proved their result for NearBipartiteness by giving a polynomial-time verifiable characterisation of the class of near-bipartite graphs of diameter 2 We use their characterisation as the starting point for our algorithm for Independent Feedback Vertex Set. our algorithm solves the decision problem but even finds a minimum independent feedback vertex set in a graph of diameter 2.
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