On Connected Resolving Decompositions in Graphs

Abstract

For an ordered k-decomposition $D = \{ G_1 ,G_2 ,...,G_k \}$ of a connected graph G and an edge e of G, the $D$-code of e is the k-tuple $c_D (e) = (d(e,G_1 ),d(e,G_2 ),...,d(e,G_k ))$ where d(e, G _i) is the distance from e to G _i. A decomposition $D$ is resolving if every two distinct edges of G have distinct $D$-codes. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dim _d (G). A resolving decomposition $D$ of G is connected if each G _i is connected for 1 ≤ i ≤ k. The minimum k for which G has a connected resolving k-decomposition is its connected decomposition number cd(G). Thus 2 ≤ dim _d (G) ≤ cd(G) ≤ m for every connected graph G of size m ≥ 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 ≤ t ≤ 1, there is a connected graph G of size m such that dim _d (G)/m = s and cd(G)/m = t.