Volkmann, Lutz; Zelinka, Bohdan Signed domatic number of a graph. (English) Zbl 1079.05071 Discrete Appl. Math. 150, No. 1-3, 261-267 (2005). Given a graph \(G=(V, E)\), the signed dominating function is a two-valued mapping \(f : V \rightarrow \{-1, 1\}\) such that, for each vertex \(v\in V\), \(\sum_{x\in N[v]} f(x) \geq 1\), where \(N[v]=N(v)\cup\{v\}\) is the closed neighborhood of \(v\). A signed dominating family on \(G\) is a set \(\{f_1, f_2, \ldots, f_d\}\) of signed dominating functions on \(G\) with the property that \(\sum_{i=1}^d f_i(x) \leq 1\) for each \(x\in V\). The maximum number of functions in a signed dominating family on \(G\), denoted by \(d_S(G)\), is the signed dominatic number of \(G\). The authors point out that \(d_S(G)\) is well defined and study its basic properties. Among others, they show that \(d_S(G)\) is an odd integer between \(1\) and the minimum degree of \(G\) plus one. They then determine \(d_S(G)\) in case \(G\) is a tree, a complete graph, a cycle, a fan, or a wheel, where a fan (wheel) is a graph obtained from a path (cycle) by adding a new vertex and edges joining it to all the vertices of the path (cycle). Reviewer: Van Bang Le (Rostock) Cited in 14 Documents MSC: 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) Keywords:signed dominating function; signed domination number PDFBibTeX XMLCite \textit{L. Volkmann} and \textit{B. Zelinka}, Discrete Appl. Math. 150, No. 1--3, 261--267 (2005; Zbl 1079.05071) Full Text: DOI EuDML References: [1] Cockayne, E. J.; Hedetniemi, S. T., Towards a theory of domination in graphs, Networks, 7, 247-261 (1977) · Zbl 0384.05051 [2] J.E. Dunbar, S.T. Hedetniemi, M.A. Henning, P.J. Slater, Signed domination in graphs, Graph Theory, Combinatorics, and Applications, vol. 1, Wiley, New York, 1995, pp. 311-322.; J.E. Dunbar, S.T. Hedetniemi, M.A. Henning, P.J. Slater, Signed domination in graphs, Graph Theory, Combinatorics, and Applications, vol. 1, Wiley, New York, 1995, pp. 311-322. · Zbl 0842.05051 [3] Favaron, O., Signed domination in regular graphs, Discrete Math., 158, 287-293 (1996) · Zbl 0861.05033 [4] Hattingh, J. H.; Henning, M. A.; Slater, P. J., On the algorithmic complexity of signed domination in graphs, Austral. J. Combin., 12, 101-112 (1995) · Zbl 0835.68089 [5] Haynes, T. W.; Hedetniemi, S. T.; Slater, P. J., Fundamentals of Domination in Graphs (1998), Marcel Dekker: Marcel Dekker New York · Zbl 0890.05002 [6] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs, Advanced Topics, Marcel Dekker, New York, 1998.; T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs, Advanced Topics, Marcel Dekker, New York, 1998. · Zbl 0883.00011 [7] Henning, M. A., Domination in regular graphs, Ars Combin., 43, 263-271 (1996) · Zbl 0881.05101 [8] Henning, M. A.; Slater, P. J., Inequalities relating domination parameters in cubic graphs, Discrete Math., 158, 87-98 (1996) · Zbl 0858.05058 [9] Rall, D. F., A fractional version of domatic number, Congr. Numer., 74, 100-106 (1990) [10] Slater, P. J.; Trees, E. L., Multi-fractional domination, J. Combin. Math. Combin. Comput., 40, 171-181 (2002) · Zbl 0995.05114 [11] Zelinka, B., Some remarks on domination in cubic graphs, Discrete Math., 158, 249-255 (1996) · Zbl 0861.05034 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.