# The Parameterized Complexity of the Equidomination Problem

## TL;DR

This paper investigates the computational complexity of the equidomination problem in graphs, showing fixed-parameter tractability for two parameterizations and providing characterizations and algorithms for recognizing equidominating graphs.

## Contribution

It introduces fixed-parameter tractability results for the Target-t and k-Equidomination problems and characterizes graphs with all induced subgraphs equidominating.

## Key findings

- Two parameterized versions are fixed-parameter tractable.
- A finite forbidden induced subgraph characterization is provided.
- A fast recognition algorithm for certain graphs is developed.

## Abstract

A graph $G=(V,E)$ is called equidominating if there exists a value $t \in \mathbb{N}$ and a weight function $\omega : V \rightarrow \mathbb{N}$ such that the total weight of a subset $D\subseteq V$ is equal to $t$ if and only if $D$ is a minimal dominating set. To decide whether or not a given graph is equidominating is referred to as the Equidomination problem. In this paper we show that two parameterized versions of the Equidomination problem are fixed-parameter tractable: the first parameterization considers the target value $t$ leading to the Target-$t$ Equidomination problem. The second parameterization allows only weights up to a value $k$, which yields the $k$-Equidomination problem. In addition, we characterize the graphs whose every induced subgraph is equidominating. We give a finite forbidden induced subgraph characterization and derive a fast recognition algorithm.

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Source: https://tomesphere.com/paper/1705.05599