### Uniquely $K_r$-Saturated Graphs

#### ##article.abstract##

A graph $G$ is

*uniquely $K_r$-saturated*if it contains no clique with $r$ vertices and if for all edges $e$ in the complement, $G+e$ has a unique clique with $r$ vertices. Previously, few examples of uniquely $K_r$-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs. We find several new uniquely $K_r$-saturated graphs with $4 \leq r \leq 7$, as well as two new infinite families based on Cayley graphs for $\mathbb{Z}_n$ with a small number of generators.#### ##article.subject##

uniquely saturated graphs; Cayley graphs; orbital branching; computational combinatorics