On the p-reinforcement and the complexity

TL;DR

This paper investigates the p-reinforcement number in graphs, providing exact values for specific graph classes, establishing upper bounds, and proving NP-hardness of the decision problem for general graphs.

Contribution

It introduces the concept of p-reinforcement, computes it for certain graphs, and proves the NP-hardness of the related decision problem.

Findings

r_p(G) determined for paths, cycles, complete t-partite graphs

Established upper bounds for r_p(G)

Decision problem for r_p(G) is NP-hard for p ≥ 2

Abstract

Let $G=(V,E)$ be a graph and $p$ be a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set if each vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number $\g_p(G)$ is the size of a smallest $p$-dominating set of $G$. The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with $\g_p(G')<\g_p(G)$. In this paper, we give an original study on the $p$-reinforcement, determine $r_p(G)$ for some graphs such as paths, cycles and complete $t$-partite graphs, and establish some upper bounds of $r_p(G)$. In particular, we show that the decision problem on $r_p(G)$ is NP-hard for a general graph $G$ and a fixed integer $p\geq 2$.