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k-Efficient partitions of graphs | ||
Communications in Combinatorics and Optimization | ||
مقاله 4، دوره 4، شماره 2، اسفند 2019، صفحه 109-122 اصل مقاله (419.76 K) | ||
نوع مقاله: Original paper | ||
شناسه دیجیتال (DOI): 10.22049/cco.2019.26446.1112 | ||
نویسندگان | ||
M Chellali1؛ Teresa W. Haynes* 2؛ Stephen T. Hedetniemi3 | ||
1LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria | ||
2Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0002 USA | ||
3Professor Emeritus, School of Computing, Clemson University, Clemson, SC 29634 USA | ||
چکیده | ||
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. | ||
کلیدواژهها | ||
domination؛ Efficient Domination؛ Distance-$k$ domination | ||
مراجع | ||
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