k-Efficient partitions of graphs

Document Type : Original paper


1 LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

2 Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0002 USA

3 Professor Emeritus, School of Computing, Clemson University, Clemson, SC 29634 USA


A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficient
dominating set if every vertex of $G$ is dominated exactly once by the
vertices 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 set
of $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices at
distance~1 from it in $G$. In this paper, we generalize the concept of
efficient domination by considering $k$-efficient domination partitions of
the vertex set of $G$, where each element of the partition is a set
consisting 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 $k$-efficient
domination number for general graphs, and for $k in {1,2}$, we give exact
values for some graph families. Complexity results are also obtained.


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