$(1,j)$-set problem in graphs

TL;DR

This paper investigates the $(1,j)$-set problem in graphs, providing bounds, complexity results, and algorithms for specific graph classes, advancing understanding of domination parameters and their computational aspects.

Contribution

It offers a probabilistic upper bound on the $(1,j)$-domination number, proves NP-completeness for chordal graphs, and develops algorithms for trees and split graphs.

Findings

Probabilistic upper bound on $oldsymbol{ ext{(1,j)}}$-domination number.

NP-completeness of the problem for chordal graphs.

Algorithms for trees and split graphs for any fixed j.

Abstract

A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as $\gamma_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$. In this paper, we first obtain an upper bound on $\gamma_{(1,j)} (G)$ using probabilistic methods, for bounded minimum and maximum degree graphs. Our bound is constructive, by the randomized algorithm of Moser and Tardos [MT10], We also show that the $(1, j)$- set problem is NP-complete for chordal graphs. Finally, we design two algorithms for finding $\gamma_{(1,j)} (G)$ of a tree and a split graph, for any fixed $j$, which answers an open question posed in [CHHM13].