Abstract
A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices so that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs. We present a polynomial time recognition algorithm of total domishold graphs, and obtain partial results towards a characterization of graphs in which the above property holds in a hereditary sense. Our characterization in the case of split graphs is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.
The research presented in this paper is work towards the Ph.D. of the first author, who is partly co-financed by the European Union through the European Social Fund. Co-financing is carried out within the framework of the Operational Programme for Human Resources Development for 2007-2013, 1. development priorities Promoting entrepreneurship and adaptability; priorities 1. 3: Scholarship Scheme. The work of the second author is supported in part by the Slovenian Research Agency, research program P1-0285 and research projects J1-4010, J1-4021 and N1-0011: GReGAS, supported in part by the European Science Foundation.
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Chiarelli, N., Milanič, M. (2013). Linear Separation of Total Dominating Sets in Graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_15
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DOI: https://doi.org/10.1007/978-3-642-45043-3_15
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