k-Efficient partitions of graphs

نویسندگان

  • M Chellali LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria
  • Teresa W. Haynes Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0002 USA
چکیده مقاله:

A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this paper, we generalize the concept ofefficient domination by considering $k$-efficient domination partitions ofthe vertex set of $G$, where each element of the partition is a setconsisting of a vertex $u_i$ and all the vertices at distance~$d_i$ from it,where $d_i in {0,1, ldots, k}$. For any integer $k geq 0$, the $k$%-efficient domination number of $G$ equals the minimum order of a $k$%-efficient partition of $G$. We determine bounds on the $k$-efficientdomination number for general graphs, and for $k in {1,2}$, we give exactvalues for some graph families. Complexity results are also obtained.

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عنوان ژورنال

دوره 4  شماره 2

صفحات  109- 122

تاریخ انتشار 2019-12-01

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