Recolouring Homomorphisms to triangle-free reflexive graphs

Abstract

For a graph $H$, the $H$-recolouring problem $\operatorname{Recol}(H)$ asks, for two given homomorphisms from a given graph $G$ to $H$, if one can get between them by a sequence of homomorphisms of $G$ to $H$ in which consecutive homomorphisms differ on only one vertex. We show that, if $G$ and $H$ are reflexive and $H$ is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related $H$-reconfiguration problem $\operatorname{Recon}(H)$ of deciding whether two given homomorphisms from a given graph $G$ to $H$ are in the same component of the Hom-graph $\operatorname{Hom}(G,H)$, can be solved in polynomial time for triangle-free reflexive graphs $H$.

Figures (4)

• Figure 1: Two homomorphisms from a long cycle $C$ to a graph $H$. The bold edges are the non-loop edges in the image of each homomorphism. The first homomorphism "wraps around" a cycle $C'$ of $H$ a bounded number of times. If $C$ is sufficiently long, then the first homomorphism can be reconfigured to the second. Triangle-freeness of $H$ makes it impossible for the image of $C$ to completely leave cycle $C'$.
• Figure 2: Two homomorphisms $\phi$ and $\psi$ from a graph $G$ to a triangle-free graph $H$. The bold edges indicate the non-loop edges in the images of the homomorphisms. There are no tight closed walks and the restrictions of these homomorphisms to any given cycle of $G$ can be reconfigured, but $\phi$ cannot be reconfigured to $\psi$.
• Figure 3: The Petersen graph and a finite portion of its universal cover.
• Figure 4: The cycle $C = C_1 \cdot_X C_2$ where $C_1 = U_1 \cdot X \cdot V_1$ and $C_2 = U_2 \cdot {X^{-1}} \cdot V_2$

Theorems & Definitions (38)

• Theorem 1.1
• Corollary 1.2
• proof
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• proof
• Lemma 3.6