Connected Joins in Graphs

A join in a graph is a set F of edges such that for every circuit C, |C ∩ F | ≤ |C \ F |. We study the problem of finding a connected join covering a given subset of vertices of the graph, that is a Steiner tree which is a join at the same time. This turns out to contain the question of finding a T -join of minimum cardinality (or weight) which is, in addition, connected. This last problem is mentioned to be open in a survey of Frank [7], and is motivated by its link to integral packings of T -cuts: if a minimum T -join F is connected, then there exists an integral packing of T -cuts of cardinality |F |. The problems we deal with are closely related to some known NPcomplete problems: deciding the existence of a connected T -join; finding the minimum cardinality of a connected T -join; the Steiner tree problem; subgraph isomorphism. We also explore some of these connections.