The Local Metric Dimension and Distance-Edge-Monitoring Number of Graph

Let $G$ be a graph and $u,v,w\in V(G)$. The vertex $w$ is said to resolve a pair $u$ and $v$ if and only if $d_G(w,u)\ne d_G(w,v)$. A set $W\subseteq V(G)$ is defined as a resolving set of $G$ if for all $u,v\in V(G)$, the pair $(u,v)$ is resolved by some $w\in W$. The minimum cardinality of a resolve set of $G$ is defined as $dim(G)$. A set $R\subseteq V(G)$ is a local resolve set of $G$ if for all $u,v\in V(G)$ such that $uv\in E(G)$, the pair $(u,v)$ is resolved by some $r\in R$. The minimum cardinality of a local resolve set of $G$ is defined as ${dim}_{\ell }(G)$. An edge $uv\in E(G)$ is said to be monitored by $x\in V(G)$ if $d_G(x,u)\ne d_{G-uv}(x,u)$ or $d_G(x,v)\ne d_{G-uv}(x,v)$. A set $M\subseteq V(G)$ is a distance-edge-monitoring (DEM) set if for all $e\in E(G)$, $e$ is monitored by some $v\in M$. The minimum cardinality of a $DEM$ set of $G$ is defined as $dem(G)$.

In this paper, we obtained that $1\le {dim}_{\ell }(G)\le dem(G)\le n-1$ for all non trivial graphs with order $n$, and the exact value of ${dim}_{\ell }(G)$ and $dem(G)$ for $G\in \{T_n,K_n,B_n,N{E}_n$, $Koch(n)\}$. Also, we obtained that if ${dim}_{\ell }(G)=1$ then $dem(G)\le \lfloor \frac{n}{2}\rfloor$. With respect to the relation between the defined graph invariants, it was proved a bound for $(dem-dim)(n)$ for $G\in {𝔊}_n=\{G||V(G)|=n\}$ and the exact values of $(dim-{dim}_{\ell })(n)$ for $G\in {𝔊}_n$, where $(dem-dim)(n)$ (resp, $(dim-{dim}_{\ell })(n)$)is the maximum value of $dem(G)-dim(G)$ (resp, $dem(G)-{dim}_{\ell }(G)$) over all graphs $G$ with order $n$. Finally, we proved that for $2\le s\le t\le s+\lfloor \frac{n-2s}{3}\rfloor$, there exists a graph with order $n$ such that ${dim}_{\ell }(G)=s$ and $dem(G)=t$.