Computational complexity aspects of super domination

TL;DR

This paper studies the super domination number in graphs, providing formulas for trees and subdivisions, proving NP-completeness in bipartite graphs, and introducing II-matching numbers with computational hardness results.

Contribution

It offers an exact formula for trees, analyzes super domination in subdivisions, and establishes NP-completeness for bipartite graphs, introducing II-matching numbers.

Findings

Exact super domination number formula for trees

NP-completeness of super domination in bipartite graphs with girth ≥ 8

Determination of super domination number for all k-subdivisions

Abstract

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super dominating set of $G$ is the super domination number of $G$. An exact formula for the super domination number of a tree $T$ is obtained and demonstrated that a smallest super dominating set of $T$ can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph $G$ is at most a given integer if $G$ is a bipartite graph of girth at least $8$. The super domination number is determined for all $k$-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determine.