5 Let C = {C n : n ≥ 5} and let L = {G : G is the line graph of an internally 4-connected cubic graph}. 6 A classical result of Martinov states that every 4-connected graph G can be constructed from graphs in 7 C ∪L by repeatedly splitting vertices. In this paper we prove that, in fact, G can be constructed from C 5 8 or C 6 in the same way, unless G belongs to C ∪L. Moreover, if G is nonplanar...

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 m...