# Article

Full entry | PDF   (0.3 MB)
Keywords:
resolving dominating set; resolving domination number
Summary:
For an ordered set $W =\lbrace w_1, w_2, \cdots , w_k\rbrace$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r(v|W) = (d(v, w_1),d(v, w_2) ,\cdots , d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $\dim G$. A set $S$ of vertices in $G$ is a dominating set for $G$ if every vertex of $G$ that is not in $S$ is adjacent to some vertex of $S$. The minimum cardinality of a dominating set is the domination number $\gamma (G)$. A set of vertices of a graph $G$ that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called the resolving domination number $\gamma _r(G)$. In this paper, we investigate the relationship among these three parameters.
References:
[1] G. Chartrand, L. Eroh, M. Johnson, O. R. Oellermann: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105 (2000), 99–113. DOI 10.1016/S0166-218X(00)00198-0 | MR 1780464
[2] G. Chartrand, L. Lesniak: Graphs & Digraphs, third edition. Chapman & Hall, New York, 1996. MR 1408678
[3] G. Chartrand, C. Poisson, P. Zhang: Resolvability and the upper dimension of graphs. International J. Comput. Math. Appl. 39 (2000), 19–28. MR 1763834
[4] T. W. Haynes, S. T. Hedetniemi, P. J. Slater: Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998. MR 1605684
[5] F. Harary, R. A. Melter: On the metric dimension of a graph. Ars Combin. 2 (1976), 191–195. MR 0457289
[6] P. J. Slater: Leaves of trees. Congr. Numer. 14 (1975), 549–559. MR 0422062 | Zbl 0316.05102
[7] P. J. Slater: Dominating and reference sets in graphs. J. Math. Phys. Sci. 22 (1988), 445–455. MR 0966610

Partner of