# Article

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Keywords:
resolving set; basis; dimension; connected resolving set; connected resolving number
Summary:
For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace$ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, 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 for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $<W>$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace (1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace$. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied.
References:
[1] P.  Buczkowski, G.  Chartrand, C.  Poisson and P. Zhang: On $k$-dimensional graphs and their bases. Period. Math. Hungar. 46 (2003), 25–36. DOI 10.1023/A:1025745406160 | MR 1975342
[2] G.  Chartrand, L. Eroh, M.  Johnson and O.  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
[3] G.  Chartrand and L.  Lesniak: Graphs $\&$ Digraphs. Third edition. Chapman $\&$ Hall, New York, 1996. MR 1408678
[4] G.  Chartrand, C. Poisson and P. Zhang: Resolvability and the upper dimension of graphs. Inter. J. Comput. Math. Appl. 39 (2000), 19–28. MR 1763834
[5] G.  Chartrand, M.  Raines and P. Zhang: The directed distance dimension of oriented graphs. Math. Bohem. 125 (2000), 155–168. MR 1768804
[6] M. R.  Garey and D. S.  Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979. MR 0519066
[7] F.  Harary and R. A.  Melter: On the metric dimension of a graph. Ars Combin. 2 (1976), 191–195. MR 0457289
[8] M. A.  Johnson: Browsable structure-activity datasets. Submitted.
[9] P. J.  Slater: Leaves of trees. Congr. Numer. 14 (1975), 549–559. MR 0422062 | Zbl 0316.05102
[10] P. J.  Slater: Dominating and reference sets in graphs. J.  Math. Phys. Sci. 22 (1988), 445–455. MR 0966610

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