# Article

Full entry | PDF   (0.3 MB)
Keywords:
distance; resolving decomposition; connected resolving decomposition
Summary:
For an ordered $k$-decomposition $\mathcal D = \lbrace G_1, G_2,\dots , G_k\rbrace$ 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 $\dim _d(G)$. 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 $2 \le \dim _d(G) \le \mathop {\mathrm cd}(G) \le m$ for every connected graph $G$ of size $m \ge 2$. All nontrivial connected graphs of size $m$ with connected decomposition number 2 or $m$ have been characterized. We present characterizations for connected graphs of size $m$ with connected decomposition number $m-1$ or $m-2$. It is shown that each pair $s, t$ of rational numbers with $0 < s \le t \le 1$, there is a connected graph $G$ of size $m$ such that $\dim _d(G)/m = s$ and $\mathop {\mathrm cd}(G) / m = t$.
References:
[1] J.  Bosak: Decompositions of Graphs. Kluwer Academic, Boston, 1990. MR 1071373 | Zbl 0701.05042
[2] G.  Chartrand, D. Erwin, M. Raines and P.  Zhang: The decomposition dimension of graphs. Graphs and Combin. 17 (2001), 599–605. DOI 10.1007/PL00007252 | MR 1876570
[3] G.  Chartrand and L.  Lesniak: Graphs & Digraphs, third edition. Chapman & Hall, New York, 1996. MR 1408678
[4] H.  Enomoto and T.  Nakamigawa: On the decomposition dimension of trees. Discrete Math. 252 (2002), 219–225. DOI 10.1016/S0012-365X(01)00454-X | MR 1907757
[5] A.  Küngen and D. B.  West: Decomposition dimension of graphs and a 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 and R. A.  Melter: On the metric dimension of a graph. Ars Combin. 2 (1976), 191–195. MR 0457289
[9] B. L.  Hulme, A. W.  Shiver and P. J.  Slater: FIRE: A subroutine for fire protection network analysis. SAND 81-1261, Sandia National Laboratories, Albuquerque, 1981.
[10] B. L.  Hulme, A. W.  Shiver and P. J.  Slater: Computing minimum cost fire protection. SAND 82-0809, Sandia National Laboratories, Albuquerque, 1982.
[11] B. L.  Hulme, A. W.  Shiver and P. J.  Slater: A Boolean algebraic analysis of fire protection. Annals of Discrete Mathematics, Algebraic Structure in Operations Research, 1984, pp. 215–228. MR 0780023
[12] P. J. Slater: Leaves of trees. Congr. Numer. 14 (1975), 549–559. MR 0422062 | Zbl 0316.05102
[13] P. J. Slater: Dominating and reference sets in graphs. J.  Math. Phys. Sci. 22 (1988), 445–455. MR 0966610
[14] V.  Saenpholphat and P.  Zhang: Connected resolving decompositions in graphs. Math. Bohem. 128 (2003), 121–136. MR 1995567

Partner of