Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284220191201k-Efficient partitions of graphs1091221385210.22049/cco.2019.26446.1112ENMChellaliLAMDA-RO Laboratory,
Department of Mathematics,
University of Blida,
B.P. 270, Blida, AlgeriaTeresa W.HaynesDepartment of Mathematics and Statistics,
East Tennessee State University,
Johnson City, TN 37614-0002 USAStephen T.HedetniemiProfessor Emeritus, School of Computing,
Clemson University,
Clemson, SC 29634 USAJournal Article20181010A 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. http://comb-opt.azaruniv.ac.ir/article_13852_478f6dfde08ed596744a7773348bc29a.pdf