# On the equality of domination number and $ 2 $-domination number

## TL;DR

This paper investigates when the 2-domination number equals the domination number in graphs, characterizes such graphs within a large class, and proves the problem's NP-hardness in the general case.

## Contribution

It characterizes graphs where the domination number equals the 2-domination number and establishes NP-hardness of deciding this equality.

## Key findings

- Characterization of graphs with equal domination and 2-domination numbers
- Proof of NP-hardness for deciding the equality in general graphs
- Necessary and sufficient conditions for hereditary equality

## Abstract

The 2-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ \gamma_2(G) $ cannot be smaller than the domination number $ \gamma(G) $. We consider a large class of graphs and characterize those members which satisfy $\gamma_2=\gamma$. For the general case, we prove that it is NP-hard to decide whether $\gamma_2=\gamma$ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.