Cockayne, E. J.; Mynhardt, C. M.; Yu, Bo Universal minimal total dominating functions in graphs. (English) Zbl 0804.90122 Networks 24, No. 2, 83-90 (1994). Summary: A total dominating function (TDF) of a graph \(G= (V,E)\) is a function \(f: V\to [0,1]\) such that for each \(v\in V\), \(\sum_{u\in N(v)}f(u)\geq 1\) [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. Cited in 1 ReviewCited in 6 Documents MSC: 90C35 Programming involving graphs or networks Keywords:total dominating function; universal MTDF in a graph; convex combinations PDF BibTeX XML Cite \textit{E. J. Cockayne} et al., Networks 24, No. 2, 83--90 (1994; Zbl 0804.90122) Full Text: DOI OpenURL References: [1] Allan, Discr. Math. 49 pp 7– (1984) [2] Bertossi, Inform. Process. Lett. 23 pp 131– (1986) [3] Chang, Oper. Res. Lett. 8 pp 53– (1989) [4] Cockayne, Networks 10 pp 211– (1980) [5] Cockayne, Ars. Comb. [6] Cockayne, J. Combinat. Math. Combin. Comput. 10 pp 23– (1991) [7] , and , Convexity of minimal dominating functions of trees. Submitted. · Zbl 0839.05030 [8] Cockayne, Discr. Math. [9] Cockayne, Bull. of ICA 5 pp 37– (1992) [10] Fricke, Congr. Numer. 77 pp 87– (1990) [11] private communication, 1990. [12] Hedetniemi, Discrete Math. 86 pp 257– (1990) [13] Lovasz, Ann. Discr. Math. 29 pp 273– (1986) [14] Convex Sets. McGraw Hill, New York (1964). [15] Convexity of minimal total dominating functions in graphs. Master’s Thesis, University of Victoria (1992). 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.