# Convexity of minimal total dominating functions in graphs

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## 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.

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