A characterisation of universal minimal total dominating functions in trees. (English) Zbl 0824.05034

A total dominating function (TDF) of a graph \(G= (V, E)\) is a mapping \(f\) of \(V\) into the closed interval \([0, 1]\) of real numbers with the property that for each \(v\in V\) the sum of values of \(f\) in all vertices adjacent to \(v\) is at least 1. A TDF is called minimal (MTDF), if there exists no TDF \(g\) of \(G\) with the property that \(g(x)\leq f(x)\) for all \(x\in V\) and \(g(x_ 0)< f(x_ 0)\) at least for one \(x_ 0\in V\). A convex combination of two functions \(f\), \(g\) is a function \(h_ \lambda\) given by \(h_ \lambda(x)= \lambda f(x)+ (1- \lambda) g(x)\), where \(\lambda\) is a real number from the interval \([0,1]\). If a MTDF has the property that its convex combination with any other MTDF is also a MTDF, it is called a universal MTDF. In the paper the universal minimal total dominating functions of trees are characterized.


05C35 Extremal problems in graph theory
05C05 Trees
05C75 Structural characterization of families of graphs
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