Efficient open domination in digraphs

Let G be a digraph. A set S ⊆ V (G) is called an efficient total dominating set if the set of open out-neighborhoods N−(v) ∈ S is a partition of V (G). We say that G is efficiently open-dominated if both G and its reverse digraph G− have an efficient total dominating set. Some properties of efficiently open dominated digraphs are presented. Special attention is given to tournaments and directed tori.

[1]  Jochen Harant,et al.  On Domination in Graphs , 2005, Discuss. Math. Graph Theory.

[2]  Quentin F. Stout,et al.  Unique Domination in Cross-Product Graphs , 1996 .

[3]  Allen J. Schwenk,et al.  Efficient dominating sets in labeled rooted oriented trees , 2005, Discret. Math..

[4]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[5]  Jun-Ming Xu,et al.  The bondage numbers and efficient dominations of vertex-transitive graphs , 2008, Discret. Math..

[6]  Jun-Ming Xu,et al.  The total domination and total bondage numbers of extended de Bruijn and Kautz digraphs , 2007, Comput. Math. Appl..

[7]  Martin Knor,et al.  Domination in a digraph and in its reverse , 2009, Discret. Appl. Math..

[8]  Ramón Beivide,et al.  Perfect Codes for Metrics Induced by Circulant Graphs , 2007, IEEE Transactions on Information Theory.

[9]  Italo J. Dejter,et al.  Efficient dominating sets in Cayley graphs , 2003, Discret. Appl. Math..

[10]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[11]  Anton Cerný,et al.  Efficient domination in directed tori and the Vizing's conjecture for directed graphs , 2009, Ars Comb..

[12]  Heather Gavlas,et al.  Efficient Open Domination , 2002, Electron. Notes Discret. Math..

[13]  Jan Kratochvíl,et al.  Perfect codes over graphs , 1986, J. Comb. Theory, Ser. B.