Breaking the Symmetries of Amenable Graphs

TL;DR

This paper explores methods to break symmetries in graphs using distinguishing labelings and fixing sets, providing efficient algorithms for computing these measures specifically for amenable graphs.

Contribution

It introduces efficient algorithms to compute the distinguishing number and fixing number for amenable graphs, based on their characterization.

Findings

Both $D(G)$ and $Fix(G)$ can be computed in $O((|V(G)|+|E(G)|) \\log |V(G)|)$ time for amenable graphs.

Color refinement effectively determines automorphisms in amenable graphs.

The paper extends understanding of symmetry breaking in graph theory.

Abstract

In this paper, we consider two ways of breaking a graph's symmetry: distinguishing labelings and fixing sets. A distinguishing labeling $\phi$ of $G$ colors the vertices of $G$ so that the only automorphism of the labeled graph $(G, \phi)$ is the identity map. The distinguishing number of $G$, $D(G)$, is the fewest number of colors needed to create a distinguishing labeling of $G$. A subset $S$ of vertices is a fixing set of $G$ if the only automorphism of $G$ that fixes every element in $S$ is the identity map. The fixing number of $G$, $Fix(G)$, is the size of a smallest fixing set. A fixing set $S$ of $G$ can be translated into a distinguishing labeling $\phi_S$ by assigning distinct colors to the vertices in $S$ and assigning another color (e.g., the ``null" color) to the vertices not in $S$. Color refinement is a well-known efficient heuristic for graph isomorphism. A graph $G$ is amenable if, for any graph $H$, color refinement correctly determines whether $G$ and $H$ are isomorphic or not. Using the characterization of amenable graphs by Arvind et al. as a starting point, we show that both $D(G)$ and $Fix(G)$ can be computed in $O((|V(G)|+|E(G)|) \log |V(G)|)$ time when $G$ is an amenable graph.