Determining a graph from its reconfiguration graph

TL;DR

The paper investigates to what extent a graph $G$ can be recovered from its recolouring reconfiguration graphs. It establishes that $G$ is determinable from the $k$-recolouring graph $\mathcal{C}_k(G)$ whenever $k>\chi(G)$, with a polynomial-time recovery that does not require knowing $k$, and strengthens the bound to $k>\min\{n,2\Delta\}$ when a finite search over $O((kn)^2)$ colourings suffices. It extends the approach to $k$-Kempe-recolouring graphs, proving determinability for $k>\chi(G)+1$ (with non-uniqueness existing at $k=\chi(G)$ via Mycielski-type constructions) and offering a polynomial-time recovery bound analogous to the vertex recolouring case. In contrast, for independent set reconfiguration, the paper shows that, aside from a few trivial cases, the graph $G$ is not reconstructible from any of the independent set reconfiguration graphs, with a detailed split across TAR, TJ, and TS rules. Overall, the work sharpens the known bounds, provides constructive recovery algorithms where possible, and delineates the boundary between identifiability and non-identifiability across recolouring models.

Abstract

Given a graph $G$ and a natural number $k$, the $k$-recolouring graph $\mathcal{C}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and whose edges link pairs of colourings which differ at exactly one vertex of $G$. Recently, Hogan et al. proved that $G$ can be determined from $\mathcal{C}_k(G)$ provided $k$ is large enough (quadratic in the number of vertices of $G$). We improve this bound by showing that $k=χ(G)+1$ colours suffice, and provide examples of families of graphs for which $k=χ(G)$ colours do not suffice. We then extend this result to $k$-Kempe-recolouring graphs, whose vertices are again the $k$-colourings of a graph $G$ and whose edges link pairs of colourings which differ by swapping the two colours in a connected component induced by selecting those two colours. We show that $k=χ(G)+2$ colours suffice to determine $G$ in this case. Finally, we investigate the case of independent set reconfiguration, proving that in only a few trivial cases is one guaranteed to be able to determine a graph $G$.

Figures (1)

• Figure 1: The graph $G_0$ with chromatic number $\chi$ from Construction \ref{['ex:nofrozen']}. Gray zones indicate complete joins.

Theorems & Definitions (42)

• Theorem 1.1: Hogan et al. hogan2024
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof
• Corollary 2.2