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For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = mind(v,x)|x ∈ S. For an ordered k-partition Π = S₁,S₂,...,Sₖ of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = S₁,S₂,...,Sₖ...

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 if the subgraph...

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 $G$ has a connected...

For a nontrivial connected graph $G$ of order $n$ and a linear ordering $s{v}_1,v_2,...,v_n$ of vertices of $G$, define $d\left(s\right)={\sum }_{i=1}^{n-1}d(v_i,v_{i+1})$. The traceable number $t\left(G\right)$ of a graph $G$ is $t\left(G\right)=min\left\{d\right(s\left)\right\}$ and the upper traceable number $t^{+}\left(G\right)$ of $G$ is $t^{+}\left(G\right)=max\left\{d\left(s\right)\right\},$ where the minimum and maximum are taken over all linear orderings $s$ of vertices of $G$. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs $G$ for which $t^{+}\left(G\right)-t\left(G\right)=1$ are characterized and a formula for the upper...

For a connected graph $G$ of order $n\ge 3$ and an ordering $s{v}_1$, $v_2,\cdots ,v_n$ of the vertices of $G$, $d\left(s\right)={\sum }_{i=1}^{n-1}d(v_i,v_{i+1})$, where $d(v_i,v_{i+1})$ is the distance between $v_i$ and $v_{i+1}$. The traceable number $t\left(G\right)$ of $G$ is defined by $t\left(G\right)=min\left\}d\left(s\right)\right\{,$ where the minimum is taken over all sequences $s$ of the elements of $V\left(G\right)$. It is shown that if $G$ is a nontrivial connected graph of order $n$ such that $l$ is the length of a longest path in $G$ and $p$ is the maximum size of a spanning linear forest in $G$, then $2n-2-p\le t\left(G\right)\le 2n-2-l$ and both these bounds are sharp. We establish a formula for the traceable number of...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathrm{i}r\left(G\right)=1$, $n-1$,...

Let $D$ be an oriented graph of order $n$ and size $m$. A $\gamma$-labeling of $D$ is a one-to-one function $fV\left(D\right)\to \{0,1,2,...,m\}$ that induces a labeling $f^{\text{'}}E\left(D\right)\to \{\pm 1,\pm 2,...,\pm m\}$ of the arcs of $D$ defined by $f^{\text{'}}\left(e\right)=f\left(v\right)-f\left(u\right)$ for each arc $e=(u,v)$ of $D$. The value of a $\gamma$-labeling $f$ is $\mathrm{v}al\left(f\right)={\sum }_{e\in E\left(G\right)}{f}^{\text{'}}\left(e\right).$ A $\gamma$-labeling of $D$ is balanced if the value of $f$ is 0. An oriented graph $D$ is balanced if $D$ has a balanced labeling. A graph $G$ is orientably balanced if $G$ has a balanced orientation. It is shown that a connected graph $G$ of order $n\ge 2$ is orientably balanced unless $G$ is a tree, $n\equiv 2\left(mod4\right)$, and every vertex of...