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A total dominating set in a graph is a set of vertices such that every vertexof the graph has a neighbor in the set. We introduce and study graphs thatadmit non-negative real weights associated to their vertices such that a set ofvertices is a total dominating set if and only if the sum of the correspondingweights exceeds a certain threshold. We show that these graphs, which we calltotal domishold graphs, form a non-hereditary class of graphs properlycontaining the classes of threshold graphs and the complements of domisholdgraphs, and are closely related to threshold Boolean functions and thresholdhypergraphs. We present a polynomial time recognition algorithm of totaldomishold graphs, and characterize graphs in which the above property holds ina hereditary sense. Our characterization is obtained by studying a new familyof hypergraphs, defined similarly as the Sperner hypergraphs, which may be ofindependent interest.

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs