Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284220191201k-Efficient partitions of graphs1091221385210.22049/cco.2019.26446.1112ENM ChellaliLAMDA-RO Laboratory,
Department of Mathematics,
University of Blida,
B.P. 270, Blida, Algeria0000-0001-5231-6195Teresa 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. https://comb-opt.azaruniv.ac.ir/article_13852_478f6dfde08ed596744a7773348bc29a.pdf