Realizability problem of distance-edge-monitoring numbers

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M, e)$ be the set of pairs $(x, y)$ with a vertex $x$ of $M$ and a vertex $y$ of $V(G)$ such that $d_{G}(x, y)\neq d_{G-e}(x, y)$. %For a vertex $x$, let $EM(x)$ be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v)\in P(\{x\}, e)$. If $e \in EM(x)$, then we say that $e$ can be monitored by the vertex $x$. Given a vertex $x$, an edge $e$ is said to be monitored by $x$ if there exists a vertex $v$ in $G$ such that $(x, v)\in P(\{x\}, e)$, and the collection of such edges is $EM(x)$. A set $M$ of vertices of a graph $G$ is distance-edge-monitoring (DEM for short) set if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The DEM number $\operatorname{dem}(G)$ of a graph $G$ is defined as the smallest size of such a set in $G$. The vertices of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the distance from $x$ to $y$ increases, and thus we are able to detect the failure. In this paper, we first give some bounds or exact values of line graphs of trees, grids, complete bipartite graphs, and obtain the exact values of DEM numbers for some graphs and their line graphs, including the friendship and wheel graphs. Next, for each $n, m>1$, we obtain that there exists a graph $G_{n,m}$ such that $\operatorname{dem}(G_{n,m})=n$ and $\operatorname{dem}(L(G_{n,m}))=4 \ or \ 2n+t$, for each integer $t\geq 0$. In the end, the DEM number for the line graph of a small world network (DURT) is given.