Abstract
A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G has an immediate successor in W. The total domination number of G is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than g(G)−1, where g(G) is the number of vertices of the smallest directed cycle contained in G. We prove that these new conjectures are equivalent to the Caccetta–Häggkvist conjecture which asserts that g(G)−1<nr in every digraph on n vertices with minimum outdegree at least r>0.
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