Total dominating functions in trees: Minimality and convexity

A total dominating function (TDF) of a graph G = (V, E) is a function f: V ← [0, 1] such that for each v ? V, Σ_u?N(v) f(u) ≥ 1 (where N(v) denotes the set of neighbors of vertex v). Convex combinations of TDFs are also TDFs. However, convex combinations of minimal TDFs (i.e., MTDFs) are not necessarily minimal. In this paper we discuss the existence in trees of a universal MTDF (i.e., an MTDF whose convex combinations with any other MTDF are also minimal). © 1995 John Wiley & Sons, Inc.