On the total $(k,r)$-domination number of random graphs

TL;DR

This paper extends bounds on the total $(k,r)$-domination number to random graphs for $k \\geq 3$ and graphs with large girth, broadening understanding of dominating sets in various graph models.

Contribution

It generalizes previous results to include larger $k$ values in random graphs and provides bounds for graphs with large girth.

Findings

Upper bounds on $\\gamma^{t}_{(k,r)}(G(n,p))$ for $k \\geq 3$ in random graphs.

Upper bounds on $\\gamma^{t}_{(k,r)}(G)$ for graphs with large girth.

Abstract

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is called the total $(k,r)$-domination number of $G$, denoted by $\gamma^{t}_{(k,r)}(G)$. We previously gave an upper bound on $\gamma^{t}_{(2,r)}(G(n,p))$ in random graphs with non-fixed $p \in (0,1)$. In this paper we generalize this result to give an upper bound on $\gamma^{t}_{(k,r)}(G(n,p))$ in random graphs with non-fixed $p \in (0,1)$ for $k\geq 3$ as well as present an upper bound on $\gamma^{t}_{(k,r)}(G)$ in graphs with large girth.