Reconfiguring homomorphisms to reflexive graphs via a simple reduction

TL;DR

A simple reduction that generalizes all known algorithmic results for H-Recoloring for square-free irreflexive graphs and yields a polynomial-time algorithm for H-Recoloring for square-free reflexive graphs.

Abstract

Given a graph $G$ and two graph homomorphisms $α$ and $β$ from $G$ to a fixed graph $H$, the problem $H$-Recoloring asks whether there is a transformation from $α$ to $β$ that changes the image of a single vertex at each step and keeps a graph homomorphism throughout. The complexity of the problem depends among other things on the presence of loops on the vertices. We provide a simple reduction that, using a known algorithmic result for $H$-Recoloring for square-free irreflexive graphs $H$, yields a polynomial-time algorithm for $H$-Recoloring for square-free reflexive graphs $H$. This generalizes all known algorithmic results for $H$-Recoloring for reflexive graphs $H$. Furthermore, the construction allows us to recover some of the known hardness results. Finally, we provide a partial inverse of the construction for bipartite instances.

Figures (2)

• Figure 1: On the left, a graph $H$ whose maximal cliques are the two triangles in blue and three edges in red. On the right, the vertex-clique incidence graph $\mathcal{K}(H)$.
• Figure 2: On the left, two reconfiguration moves starting from the homomorphisms $G \to H$ of Figure \ref{['fig:K(H)']}. On the right, the associated reconfiguration moves of homomorphisms $\mathcal{E}(G) \to \mathcal{K}(H)$. Notice that before $w$ moves, it is required to change the position of the edge $(vw)$ in $\mathcal{K}(H)$ first (since the edges $(vu)$ and $(vw)$ remain on same blue triangle after $v$ moves, this requires no change in $\mathcal{K}(H)$).

Theorems & Definitions (13)

• Theorem 1: Wrochna:20
• Conjecture 2
• Theorem 3
• Corollary 4
• Corollary 5: Lee:21
• Proposition 6
• Lemma 9: see Lee:21
• Definition 10
• Proposition 11
• proof