Spanning Trees Whose Reducible Stems Have a Few Branch Vertices

Let T be a tree. Then a vertex of T with degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. The set of leaves of T is denoted by L(T) and the set of branch vertices of T is denoted by B(T). For two distinct vertices u, v of T, let P-T[u, v] denote the unique path in T connecting u and v. Let T be a tree with B(T) not equal null . For each leaf x of T, let y(x) denote the nearest branch vertex to x. We delete V(P-T[x, y(x)]) {y(x)} from T for all x is an element of L(T). The resulting subtree of T is called the reducible stem of T and denoted by R_Stem(T). We give sharp sufficient conditions on the degree sum for a graph to have a spanning tree whose reducible stem has a few branch vertices.