Abstract A total dominating function (TDF) of a graph G = (V, E) is a function f: V → [0, 1] such that for each ν ϵ V, ΣuϵN(v) f(u) ⩾ 1 [where N(v) denotes the open neighborhood of vertex v]. Integer‐valued TDFs are precisely characteristic functions of total dominating sets of G. Convex combinations of two TDFs are themselves TDFs but convex combinations of minimal TDFs (MTDFs) are not necessarily minimal. This paper is concerned with the existence of a universal MTDF in a graph, i.e., a MTDF g such that convex combinations of g and any other MTDF are themselves minimal. © 1994 by John Wiley & Sons, Inc.