Previous |  Up |  Next


distance; resolving decomposition; connected resolving decomposition
For an ordered $k$-decomposition ${\mathcal D}= \lbrace G_1, G_2,\ldots , G_k\rbrace $ of a connected graph $G$ and an edge $e$ of $G$, the ${\mathcal D}$-code of $e$ is the $k$-tuple $c_{{\mathcal D}}(e)$ = ($d(e, G_1),$ $d(e, G_2),$ $\ldots ,$ $d(e, G_k)$), where $d(e, G_i)$ is the distance from $e$ to $G_i$. A decomposition ${\mathcal D}$ is resolving if every two distinct edges of $G$ have distinct ${\mathcal D}$-codes. The minimum $k$ for which $G$ has a resolving $k$-decomposition is its decomposition dimension $\dim _{\text{d}}(G)$. A resolving decomposition ${\mathcal D}$ of $G$ is connected if each $G_i$ is connected for $1 \le i \le k$. The minimum $k$ for which $G$ has a connected resolving $k$-decomposition is its connected decomposition number $\mathop {\mathrm cd}(G)$. Thus $2 \le \dim _{\text{d}}(G) \le \mathop {\mathrm cd}(G) \le m$ for every connected graph $G$ of size $m \ge 2$. All nontrivial connected graphs with connected decomposition number $2$ or $m$ are characterized. We provide bounds for the connected decomposition number of a connected graph in terms of its size, diameter, girth, and other parameters. A formula for the connected decomposition number of a nonpath tree is established. It is shown that, for every pair $a, b$ of integers with $3 \le a \le b$, there exists a connected graph $G$ with $\dim _{\text{d}}(G) = a$ and $\mathop {\mathrm cd}(G) = b$.
[1] J. Bosák: Decompositions of Graphs. Kluwer Academic, Boston, 1990. MR 1071373
[2] G. Chartrand, D. Erwin, M. Raines, P. Zhang: The decomposition dimension of graphs. Graphs Combin. 17 (2001), 599–605. DOI 10.1007/PL00007252 | MR 1876570
[3] G. Chartrand, L. Lesniak: Graphs $\&$ Digraphs, third edition. Chapman $\&$ Hall, New York, 1996. MR 1408678
[4] H. Enomoto, T. Nakamigawa: On the decomposition dimension of trees. Preprint. MR 1907757
[5] A. Küngen, D. B. West: Decomposition dimension of graphs and union-free family of sets. Preprint.
[6] M. A. Johnson: Structure-activity maps for visualizing the graph variables arising in drug design. J. Biopharm. Statist. 3 (1993), 203–236. DOI 10.1080/10543409308835060 | Zbl 0800.92106
[7] M. A. Johnson: Browsable structure-activity datasets. Preprint.
[8] F. Harary, R. A. Melter: On the metric dimension of a graph. Ars Combin. 2 (1976), 191–195. MR 0457289
[9] P. J. Slater: Leaves of trees. Congress. 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
Partner of
EuDML logo