# Generalized power domination in claw-free regular graphs

## TL;DR

This paper provides counterexamples to a conjecture on k-power domination in regular graphs and establishes tight bounds for k-power domination in connected claw-free regular graphs.

## Contribution

It disproves a previous conjecture and derives new tight bounds for k-power domination in claw-free regular graphs.

## Key findings

- Counterexamples negate the conjecture on k-power domination.
- Bound of n/(k+l+2) for k-power domination in certain claw-free regular graphs.
- The bounds are proven to be tight.

## Abstract

In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]). Furthermore, we focus on the study of $k$-power domination of claw-free graphs. We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most $\frac{n}{k+l+2}$, and this bound is tight.