Breaking the Symmetries of Amenable Graphs

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 $O((\mathrm{\mid }V(G)\mathrm{\mid }+\mathrm{\mid }E(G)\mathrm{\mid })\log \mathrm{\mid }V(G)\mathrm{\mid })$ time when $G$ is an amenable graph.