The distance-edge-monitoring numbers of subdivision graphs

Abstract

For a vertex set $M$ and an edge $e$ of a connected graph $G$, let $P(M,e)$ be the set of pairs $(x,y)$ with $x\in M$ and $y\in V\left(G\right)$ such that $d_G(x,y)\ne d_{G-e}(x,y)$. For a vertex $x$, let $EM\left(x\right)$ be the set of edges $e$ such that there exists a vertex $v$ satisfying $(x,v)\in P(\left\{x\right\},e)$ in $G$. A set $M$ of vertices of a graph $G$ is called a distance-edge-monitoring (DEM for short) set if, for every edge $e$ of $G$, there exist vertices $x\in M$ and $y\in V\left(G\right)$ such that $e$ belongs to all shortest paths between $x$ and $y$. We denote by $dem\left(G\right)$ the size of a smallest DEM set of $G$. Given a graph $G$, subdividing one edge $uv$ means removing the edge $uv$, adding an extra vertex $w$, and adding the edges $uw$ and $wv$. The subdivision graph of $G$, denoted by $S\left(G\right)$, is obtained from the graph $G$ by subdividing all its edges. In this paper, we study the DEM numbers of subdivision graphs for various classes of graphs including trees, cycles, complete graphs, grids and friendship graphs. We show that for every positive integers $a,b$ with $1<a\le b$, there exists a graph $G$ such that $dem\left(S\left(G\right)\right)=a$ and $dem\left(G\right)=b$. Furthermore, we obtain sharp upper and lower bounds on $\left|EM\left(x\right)\right|$ and $dem\left(S\left(G\right)\right)$ for every $x\in V\left(S\left(G\right)\right)$. We also give sufficient conditions under which $S\left(G\right)$ achieves specific values of $dem\left(S\left(G\right)\right)$ that are close to one of the bounds.