Independent Feedback Vertex Sets for Graphs of Bounded Diameter

TL;DR

This paper investigates the computational complexity of the Near-Bipartiteness problem, showing NP-completeness for graphs of diameter 3 and polynomial-time solvability for minimum independent feedback vertex sets in diameter 2 graphs.

Contribution

It resolves an open problem by proving NP-completeness for diameter 3 graphs and extends results by providing polynomial algorithms for diameter 2 graphs.

Findings

NP-complete for diameter 3 graphs

Polynomial-time solvable for diameter 2 graphs

Generalizes previous diameter 2 results

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.