Convexity of minimal total dominating functions in graphs

Abstract

A total dominating function (TDF) of a graph G=(V, E) is a function f ∶ V → [0,1] such that for each v ∈ V, the sum of f values over all neighbours of ν (i.e., all vertices adjacent to v) is at least one. Integer-valued TDFs are precisely the characteristic functions of total dominating sets of G. A minimal TDF (MTDF) is one such that decreasing any value of it makes it non-TDF. An MTDF f is called universal if convex combinations of f and any other MTDF are minimal. We give a sufficient condition for an MTDF to be universal which generalizes previous results. Also we define a splitting operation on a graph G as follows: take any vertex ν in G and a vertex ω not in G and join ω with all the neighbours of v. A graph G has a universal MTDF if and only if the graph obtained by splitting G has a universal MTDF. A corollary is that graphs obtained by the operation from paths, cycles, complete graphs, wheels, and caterpillar graphs have a universal MTDF.