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Semidomatic numbers of directed graphs. (English) Zbl 0602.05038
E. J. Cockayne and S. T. Hedetniemi have introduced the domatic number of an undirected graph. Here its variants for directed graphs are studied. An inside-dominating (or outside-dominating) set in a directed graph G is a subset D of the vertex set V(G) of G such that for each vertex \(x\in V(G)-D\) there exists a vertex \(y\in D\) such that the edge xy (or yx respectively) exists in G. A dominating set in G is a set which is simultaneously inside-semidomatic and outside-semidomatic. The maximum number of classes of a partition of V(G) into dominating (or inside- semidominating, or outside-semidominating) sets is the domatic number (or inside-semidomatic number, or outside-semidomatic number respectively) of G. Fundamental properties of these concepts are described. A special attention is paid to tournaments.

MSC:
05C35 Extremal problems in graph theory
05C20 Directed graphs (digraphs), tournaments
05C99 Graph theory
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References:
[1] COCKAYNE E. J., HEDETNIEMI S. T.: Towards a theory of domination in graphs. Networks 7, 1977, 247-261. · Zbl 0384.05051
[2] ZELINKA B.: Domatic numbers of directed graphs. Czech. Math. J. · Zbl 0847.05063
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