The Decomposition Dimension of Graphs

Abstract

For an ordered k-decomposition D fG 1 ; G 2 ; . . . ; G k g of a connected graph G and an edge e of G, the D-representation of e is the k-tuple rejD de; G 1 ; de; G 2 ; . . . ; de; G k , where de; G i is the distance from e to G i . A decomposition D is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n ! 2, there exists a tree T of order n with decT k. It is also shown that decG n for every graph G of order n ! 3 and that decK n 2n 5=3 for n ! 3.

Conditional resolvability in graphs: a survey

“…The decomposition dimensions of trees that are not paths have been studied in [7,26], where bounds for them have been determined. However, there is no general formula for the decomposition dimension of a tree that is not a path.…”

Probabilistic Methods for Decomposition Dimension of Graphs

“…For the complete graph K n , the trivial lower bound resulting from (1) is about 2 lg n−lg lg n. In [2], it was proved that dec(K n ) ≤ (2n+5)/3 for n ≥ 3. This bound was improved by Nakamigawa [8] …”

“…A decomposition is distinguishing if the resulting distance vectors for edges are distinct. Chartrand, Erwin, Raines, and Zhang [2] defined the decomposition dimension dec(G) to be the minimum number of colors in a a distinguishing decomposition of G. Since every distance vector has exactly one 0 and the remaining entries are at most d + 1 when G has diameter d, they observed that (1) dec(G) = k implies |E(G)| ≤ k(d + 1) k−1 .…”

“…Enomoto [4] proved that dec(G) ≤ n − 1 for every n-vertex graph G except complete graphs with 3 or 4 vertices. Enomoto and Nakamigawa [5] proved another lower bound in terms of the maximum degree ∆(G): (2) dec(G) ≥ lg ∆(G) + 1, where lg denotes log 2 (we also use ln for log e ). They proved that for every r ≥ 1 and k ≥ 3 there is a tree T with ∆(T ) = k and dec(T ) = lg k + r. For graphs with small diameter, the bound in (1) is sharper, and we will study graphs with this property.…”

“…These results suggest that regularity or symmetry could be relevant structural properties. Indeed, the lower bound in (2) implies that a family cannot be wellbehaved if lg ∆(G) lg diam(G)/ lg e(G) is unbounded for graphs in the family, where e(G) and diam(G) denote the number of edges and the diameter of a graph G, respectively. The family of stars shows that boundedness of this ratio is not sufficient to make a family well-behaved.…”