×

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.

MSC:

05C35 Extremal problems in graph theory
05C05 Trees
05C75 Structural characterization of families of graphs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Cockayne, E.J.; Mynhardt, C.M.; Yu, B., Universal minimal total dominating functions in graphs, Networks, 24, 83-90, (1994) · Zbl 0804.90122
[2] E.J. Cockayne, C.M. Mynhardt and B. Yu, Total dominating functions in trees: minimality and convexity, J. Graph Theory, to appear. · Zbl 0819.05035
[3] A. Stacey, Universal minimal total dominating functions of trees, preprint. · Zbl 0824.05036
[4] Yu, B., Convexity of minimal total dominating functions in graphs, () · Zbl 0876.05048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.