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Odd Cycle Transversal on $P_5$-free Graphs in Polynomial Time

Akanksha Agrawal, Paloma T. Lima, Daniel Lokshtanov, Pawel Rzążewski, Saket Saurabh, Roohani Sharma

Abstract

An independent set in a graph G is a set of pairwise non-adjacent vertices. A graph $G$ is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph $G$ along with a weight function $w$ associating a rational weight with each vertex, and the task is to find a smallest weight vertex subset $S$ in $G$ such that $G - S$ is bipartite; the weight of $S$, $w(S) = \sum_{v\in S} w(v)$. We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding $P_5$ (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on $P_4$-free graphs and NP-hard on $P_6$-free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rzążewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on $P_5$-free graphs as an open problem, this was later re-stated by Rzążewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time $n^{O(\sqrt{n})}$.

Odd Cycle Transversal on $P_5$-free Graphs in Polynomial Time

Abstract

An independent set in a graph G is a set of pairwise non-adjacent vertices. A graph is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph along with a weight function associating a rational weight with each vertex, and the task is to find a smallest weight vertex subset in such that is bipartite; the weight of , . We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on -free graphs and NP-hard on -free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rzążewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on -free graphs as an open problem, this was later re-stated by Rzążewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time .
Paper Structure (5 sections, 9 theorems, 3 figures, 1 algorithm)

This paper contains 5 sections, 9 theorems, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finding a small covering family of the connected components of a solution
  4. Reduction to Maximum Weight Independent Set on $P_5$-free Graphs and Proof of Theorem \ref{['thm:oct']}
  5. Conclusion

Key Result

Theorem 1

Odd Cycle Transversal on $P_5$-free graphs is polynomial-time solvable.

Figures (3)

  • Figure 1: Illustration of various sets and connected components used in the algorithm. The striped ovals denote removal of the corresponding vertices by the algorithm.
  • Figure 2: Illustration of various elements in the proof of Claim \ref{['claim:no-across-edges']}.
  • Figure 3: Illustration of various elements in the proof of Claim \ref{['claim:module']}.

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 1
  • Proposition 2.1: Theorem $8$, bacso1990dominating
  • Lemma 1
  • Proposition 3.1: DBLP:conf/soda/LokshantovVV14, Independent Set on $P_5$-free
  • Lemma 3
  • proof
  • Claim 3.1
  • proof
  • Claim 3.2
  • ...and 8 more