Reconfiguration of graph minors

TL;DR

This paper studies the reconfiguration of graph minors through edge contractions, analyzing when the space of all models is connected, and characterizes conditions for connectivity for specific target graphs.

Contribution

It introduces a framework for reconfiguring graph minors via edge contractions and characterizes connectivity conditions for various target graphs, including a full characterization for $K_2$.

Findings

Connectivity of the reconfiguration graph depends on properties of G and H.

Operations on G or H can preserve reconfiguration connectivity.

Complete characterization of G for $K_2$-models connectivity.

Abstract

Under the reconfiguration framework, we consider the various ways that a target graph $H$ is a {\em minor} of a host graph $G$, where a subgraph of $G$ can be transformed into $H$ by means of {\em edge contraction} (replacement of both endpoints of an edge by a new vertex adjacent to any vertex adjacent to either endpoint). Equivalently, an {\em $H$-model} of $G$ is a labeling of the vertices of $G$ with the vertices of $H$, where the contraction of all edges between identically-labeled vertices results in a graph containing representations of all edges in $H$. We explore the properties of $G$ and $H$ that result in a connected {\em reconfiguration graph}, in which nodes represent $H$-models and two nodes are adjacent if their corresponding $H$-models differ by the label of a single vertex of $G$. Various operations on $G$ or $H$ are shown to preserve connectivity. In addition, we demonstrate properties of graphs $G$ that result in connectivity for the target graphs $K_2$, $K_3$, and $K_4$, including a full characterization of graphs $G$ that result in connectivity for $K_2$-models, as well as the relationship between connectivity of $G$ and other $H$-models.