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...

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...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the 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 for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $dim\left(G\right)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected...

By a ternary structure we mean an ordered pair $(X_0,T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0,T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if $$\{x\in X_0{T}_0(u,x,v)\}\cup \{x\in X_0{T}_0(v,x,u)\}=\{u,v\}.$$ A ternary structure $(X_0,T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u,x,y\in X_0$: $T_0(x,u,y)$ if and only if $u$ belongs to an...

An edge $e$ of a $k$-connected graph $G$ is said to be $k$-removable if $G-e$ is still $k$-connected. A subgraph $H$ of a $k$-connected graph is said to be $k$-contractible if its contraction results still in a $k$-connected graph. A $k$-connected graph with neither removable edge nor contractible subgraph is said to be minor minimally $k$-connected. In this paper, we show that there is a contractible subgraph in a $5$-connected graph which contains a vertex who is not contained in any triangles. Hence, every vertex...