Perturbation results for distance-edge-monitoring numbers

TL;DR

The paper investigates how the distance-edge-monitoring number $\operatorname{dem}(G)$ behaves under perturbations. It proves a tight bound $\operatorname{dem}(G-e)-\operatorname{dem}(G)\le 2$ for edge deletions, and constructs graphs where vertex deletions can cause arbitrarily large changes, revealing nonmonotonic behavior under subgraph relations. It also establishes existence results for graphs with prescribed $\operatorname{dem}$ values and analyzes $\operatorname{dem}$ for several known graphs, complemented by an algorithm to check if a fixed DEM set remains valid after an edge deletion. These results enhance understanding of the robustness of distance-edge-monitoring in networks and provide practical tools for resilience analysis in graph-structured systems.

Abstract

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph $G=(V(G), E(G))$, a set $M \subseteq V(G)$ is a distance-edge-monitoring set if for every edge $e \in E(G)$, there is a vertex $x \in M$ and a vertex $y \in V(G)$ such that the edge $e$ belongs to all shortest paths between $x$ and $y$. The smallest size of such a set in $G$ is denoted by $\operatorname{dem}(G)$. Denoted by $G-e$ (resp. $G \backslash u$) the subgraph of $G$ obtained by removing the edge $e$ from $G$ (resp. a vertex $u$ together with all its incident edges from $G$). In this paper, we first show that $\operatorname{dem}(G-e)- \operatorname{dem}(G)\leq 2$ for any graph $G$ and edge $e \in E(G)$. Moreover, the bound is sharp. Next, we construct two graphs $G$ and $H$ to show that $\operatorname{dem}(G)-\operatorname{dem}(G\setminus u)$ and $\operatorname{dem}(H\setminus v)-\operatorname{dem}(H)$ can be arbitrarily large, where $u \in V(G)$ and $v \in V(H)$. We also study the relation between $\operatorname{dem}(H)$ and $\operatorname{dem}(G)$, where $H$ is a subgraph of $G$. In the end, we give an algorithm to judge whether the distance-edge monitoring set still remain in the resulting graph when any edge of the graph $G$ is deleted.

Figures (3)

• Figure 1: The blue edges are those of trees $T_1$ and $T_2$ in $K_4$.
• Figure 2: The graph $K{(7, 12)}$
• Figure 7: The conical graph $C(3,8)$

Theorems & Definitions (41)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 1.10