For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $dimG$. A set $S$ of vertices in $G$ is a dominating...

For an ordered $k$-decomposition $𝒟=\{G_1,G_2,\cdots ,G_k\}$ of a connected graph $G$ and an edge $e$ of $G$, the $𝒟$-code of $e$ is the $k$-tuple $c_{𝒟}\left(e\right)=(d(e,G_1),d(e,G_2),...,d(e,G_k))$, 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 ${dim}_d\left(G\right)$. A resolving decomposition $𝒟$ 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...

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 $c_{𝒟}\left(e\right)$ = ($d(e,G_1),$ $d(e,G_2),$ $...,$ $d(e,G_k)$), 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 ${dim}_{\text{d}}\left(G\right)$. A resolving decomposition $𝒟$ of $G$ is connected if each $G_i$ is connected for $1\le i\le k$. The minimum $k$ for which $G$...

An edge $e$ of a $k$-connected graph $G$ is said to be $k$-contractible (or simply contractible) if the graph obtained from $G$ by contracting $e$ (i.e., deleting $e$ and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still $k$-connected. In 2002, Kawarabayashi proved that for any odd integer $k\ge 5$, if $G$ is a $k$-connected graph and $G$ contains no subgraph $D=K_1+(K_2\cup K_{1,2})$, then $G$ has a $k$-contractible edge. In this paper, by generalizing this result, we prove that...