On the resolutions of the edge ideals of graphs

The so-called bridge-friendliness is a set of divisibility conditions on the minimal generators of monomial ideals. It was introduced by Chau and Kara as a sufficient criterion for the existence of a cellular minimal graded free resolution. It is fulfilled by large classes of monomial ideals, in particular by the edge ideals of acyclic graphs. We present a construction that, given a pair of graphs with bridge-friendly edge ideals, produces a new graph with the same property. An additional assumption is that the starting graphs both have at least one leaf.