Uniquely Kr-saturated graphs

Abstract

A graph G is uniquely Kr-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 Kr-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 Kr-saturated graphs with 4 ≤ r ≤ 7, as well as two new infinite families based on Cayley graphs for ℤn with a small number of generators.