Graph Cover-Saturation

Graph G is F -saturated if G contains no copy of graph F but any edge added to G produces at least one copy of F . One common variant of saturation is to remove the former restriction: G is F-semi-saturated if any edge added to G produces at least one new copy of F . In this paper we take this idea one step further. Rather than just allowing edges of G to be in a copy of F , we require it: G is F -covered if every edge of G is in a copy of F . It turns out that there is smooth interaction between coverage and semi-saturation, which opens for investigation a natural analogue to saturation numbers. Therefore we present preliminary cover-saturation theory and structural bounds for the cover-saturation numbers of graphs. We also establish asymptotic cover-saturation densities for cliques and paths, and upper and lower bounds (with small gaps) for cycles and stars.