About kings and dominating sets in tournaments
Résumé
In a digraph, a king is a vertex that can reach any other vertex by a directed path of length at most 2, and a dominating set is a set of vertices containing an in-neighbor of any vertex outside it. The domination number of a digraph is the cardinality of a smallest dominating set. E. Szekeres and G. Szekeres [Math. Gaz. 49 (1965), 290-293] proved that the domination number of an n-tournament is at most log 2 (n)log 2 (log 2 (n)) + 2. In this paper the concept of kings is used in order to improve this bound. Moreover, we show that for every two integers k ≥ 3 and s ≥ 2, and for all n ≥ 3s + k -2, there exists an n-tournament T which has exactly k kings and δ -(T ) = s. Furthermore, we characterize the tournaments with exactly three kings. We also treat the new concept of king degree introduced by El Sahili [Seminars on Graph Theory, 2021]; that is, in a digraph, the king degree of a vertex x, denoted by k + (x), is the number of vertices that can be reached from x by a directed path of length at most 2. The main result in this context is the characterization of the set of integers that can be the set of king degrees of the vertices of a tournament.
Domaines
Combinatoire [math.CO]Origine | Fichiers éditeurs autorisés sur une archive ouverte |
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