---
abstract: |
    A graph $G=(V,E)$ is called equidominating if there exists a value $t\in{\ensuremath{\mathbb{N}}}$ and a weight function ${\omega}\colon V \rightarrow {\ensuremath{\mathbb{N}}}$ 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 <span style="font-variant:small-caps;">Equidomination</span> problem.

    In this paper we show that two parameterized versions of the <span style="font-variant:small-caps;">Equidomination</span> problem are fixed-parameter tractable: the first parameterization considers the target value $t$ leading to the <span style="font-variant:small-caps;">Target</span>-$t$ <span style="font-variant:small-caps;">Equidomination</span> problem. The second parameterization allows only weights up to a value $k$, which yields the $k$-<span style="font-variant:small-caps;">Equidomination</span> 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.\
    **Keywords:** minimal dominating set equidominating graph kernelization parameterized complexity hereditary property
author:
- 'Oliver Schaudt[^1]'
- 'Fabian Senger[^2]'
title: The Parameterized Complexity of the Equidomination Problem
---

Introduction
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Let $G$ be a simple, undirected graph. A subset $S$ of the vertices of $G$ is called a *dominating set*, if every vertex of $G$ is an element of $S$ 