Reconfiguring Graph Homomorphisms on the Sphere

TL;DR

The first graphs for which H-Recolouring is known to be PSPACE-complete for reflexive instances are provided, which include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4-chromatic generalized Mycielski graphs, which it was not.

Abstract

Given a loop-free graph $H$, the reconfiguration problem for homomorphisms to $H$ (also called $H$-colourings) asks: given two $H$-colourings $f$ of $g$ of a graph $G$, is it possible to transform $f$ into $g$ by a sequence of single-vertex colour changes such that every intermediate mapping is an $H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs $H$ (e.g. all $C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever $H$ is a $K_{2,3}$-free quadrangulation of the $2$-sphere (equivalently, the plane) which is not a $4$-cycle. From this result, we deduce an analogous statement for non-bipartite $K_{2,3}$-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4$-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs $G$ and $H$ with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for $H$-colourings is PSPACE-complete whenever $H$ is a reflexive $K_{4}$-free triangulation of the $2$-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which $H$-Recolouring is known to be PSPACE-complete for reflexive instances.

Figures (9)

• Figure 1: The local structure near vertex $0$.
• Figure 2: An illustration of the way in which two adjacent vertices $u,v$ of $G$ are represented in the graph $G'$. Each dashed line connects signal vertices of a pair of not-both-one gadgets and each thick dotted curve encloses the four signal vertices of a not-all-zero gadget.
• Figure 3: The subgraph of $H$ induced by the vertex $0$ and its neighbours in the case $k=5$.
• Figure 4: Graphs $P$ and $L$ motivating $\Phi$ of Defintion \ref{['PhiQuad']}.
• Figure 5: An illustration of the not-all-zero gadget. Thin solid black lines represent edges, thick solid black closed curves represent the copy $H^*$ of $H$ (drawn twice for clarity), red dashed lines connect signal vertices of $\{(1,2),(0,2),(1,0)\}$-gadgets, blue dashed lines connect signal vertices of $\{(1,3),(0,3),(1,0)\}$-gadgets and the black dashed line connects signal vertices of a $\{(1,0),(0,1),(1,1)\}$-gadget.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5