On domination number of Cartesian product of directed paths

  • Authors:

  • Juan Liu;Xindong Zhang;Jixiang Meng

  • Affiliations:

  • College of Mathematics and System Sciences, Xinjiang University, Urumqi, P.R. China 830046 and College of Mathematics Sciences, Xinjiang Normal University, Urumqi, P.R. China 830054;College of Mathematics Sciences, Xinjiang Normal University, Urumqi, P.R. China 830054;College of Mathematics and System Sciences, Xinjiang University, Urumqi, P.R. China 830046

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2011

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Abstract

Let 驴(G) denote the domination number of a digraph G and let P m 驴P n denote the Cartesian product of P m and P n , the directed paths of length m and n. In this paper, we give a lower and upper bound for 驴(P m 驴P n ). Furthermore, we obtain a necessary and sufficient condition for P m 驴P n to have efficient dominating set, and determine the exact values: 驴(P 2驴P n )=n, $\gamma(P_{3}\square P_{n})=n+\lceil\frac{n}{4}\rceil$ , $\gamma(P_{4}\square P_{n})=n+\lceil\frac{2n}{3}\rceil$ , 驴(P 5驴P n )=2n+1 and $\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil$ .