Linear programming of monitoring the links of a fractional weighted network using distance

In 2022, Foucaud et al. initiated the study of a new graph-theoretic concept called distance-edge-monitoring number (DEM number for short), in the area of network monitoring. In this paper, we study linear programming for the distance-edge-monitoring problem. For a connected graph $G=\left(V\right(G),E(G\left)\right)$, let h be a function that assigns to each vertex $v\in V\left(G\right)$ a number in $[0,1]$. For a vertex subset $X\subseteq V\left(G\right)$, denote $h\left(X\right)={\Sigma }_{v\in X}h\left(v\right)$. Then, the function h is called a monitoring function of G, if for any edge e in G, the weights of these vertices which monitoring the edge e are at least 1. The fractional distance-edge-monitoring number (FDEM number for short) of G, denoted by ${\mathrm{dem}}_f\left(G\right)$, is given by ${\mathrm{dem}}_f\left(G\right)=\min \left\{H\right|h\text{is a monitoring function of}G\}$, where $H=h\left(V\right(G\left)\right)$. In this paper, we obtain some bounds or exact values for the FDEM number of some specific graphs or networks. Moreover, we study the graphs where the FDEM number equals to the DEM number. Finally, we investigate the FDEM number of some convex polytopes.