Reconfiguring dominating sets in minor-closed graph classes

TL;DR

This paper investigates the reconfiguration of dominating sets in graphs, showing that in some graphs large changes are necessary for reconfiguration, while in others, especially those with certain separator properties, smaller changes suffice.

Contribution

It establishes bounds on the size of intermediate dominating sets needed for reconfiguration in various graph classes, including toroidal graphs and hereditary classes with balanced separators.

Findings

Reconfiguration in toroidal graphs requires large intermediate dominating sets.

Hereditary classes with balanced separators allow smaller reconfiguration steps.

Bounds depend on graph class and separator properties.

Abstract

For a graph $G$, two dominating sets $D$ and $D'$ in $G$, and a non-negative integer $k$, the set $D$ is said to $k$-transform to $D'$ if there is a sequence $D_0,\ldots,D_\ell$ of dominating sets in $G$ such that $D=D_0$, $D'=D_\ell$, $|D_i|\leq k$ for every $i\in \{ 0,1,\ldots,\ell\}$, and $D_i$ arises from $D_{i-1}$ by adding or removing one vertex for every $i\in \{ 1,\ldots,\ell\}$. We prove that there is some positive constant $c$ and there are toroidal graphs $G$ of arbitrarily large order $n$, and two minimum dominating sets $D$ and $D'$ in $G$ such that $D$ $k$-transforms to $D'$ only if $k\geq \max\{ |D|,|D'|\}+c\sqrt{n}$. Conversely, for every hereditary class ${\cal G}$ that has balanced separators of order $n\mapsto n^\alpha$ for some $\alpha<1$, we prove that there is some positive constant $C$ such that, if $G$ is a graph in ${\cal G}$ of order $n$, and $D$ and $D'$ are two dominating sets in $G$, then $D$ $k$-transforms to $D'$ for $k=\max\{ |D|,|D'|\}+\lfloor Cn^\alpha\rfloor$.