Power domination in regular claw-free graphs

TL;DR

This paper establishes a tight upper bound for the power domination number in connected 4-regular claw-free graphs and introduces a linear-time algorithm for weighted power domination in trees.

Contribution

It disproves a previous conjecture and provides a new bound for a specific class of graphs, along with an efficient algorithm for trees.

Findings

Bound of (n+1)/5 for power domination in 4-regular claw-free graphs

Disproof of a prior conjecture on power domination

Linear-time algorithm for weighted power domination in trees

Abstract

In this paper, we first show that the power domination number of a connected $4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.