Anti-Path Cover on Sparse Graph Classes

TL;DR

This paper develops fixed-parameter tractable algorithms for the anti-path cover problem on sparse graph classes by leveraging Bondy-Chvatal closure and properties like tree-width and neighborhood diversity.

Contribution

It introduces a novel approach using Bondy-Chvatal closure to bound neighborhood diversity in complements of graphs with bounded tree-width, enabling efficient algorithms.

Findings

FPT algorithm for anti-path cover on certain graph classes

Bounded neighborhood diversity in complements with specific properties

Simpler proof using tree-depth instead of tree-width

Abstract

We show that it is possible to use Bondy-Chvatal closure to design an FPT algorithm that decides whether or not it is possible to cover vertices of an input graph by at most k vertex disjoint paths in the complement of the input graph. More precisely, we show that if a graph has tree-width at most w and its complement is closed under Bondy-Chvatal closure, then it is possible to bound neighborhood diversity of the complement by a function of w only. A simpler proof where tree-depth is used instead of tree-width is also presented.