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distance; resolving decomposition; connected resolving decomposition
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 $\mathop {\mathrm cd}(G) / m = t$.
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