Reducing the domination number of $P_3+kP_2$-free graphs via one edge contraction

TL;DR

This paper investigates whether a single edge contraction can reduce the domination number in connected graphs, providing polynomial-time solutions for a specific class of $P_3+kP_2$-free graphs, thus contributing to the complexity classification of the problem.

Contribution

It establishes that reducing the domination number via one edge contraction is polynomial-time solvable on $P_3+kP_2$-free graphs, extending the understanding of the problem's complexity.

Findings

Polynomial-time algorithm for $P_3+kP_2$-free graphs

Complexity dichotomy for the problem on $H$-free graphs

Extension of previous results to broader graph classes

Abstract

In this note, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by using only one edge contraction? We show that the problem is polynomial-time solvable on $P_3+kP_2$-free graphs for any $k \geq 0$ which combined with results of [1,2] leads to a complexity dichotomy of the problem on $H$-free graphs.