Bounds and extremal graphs for monitoring edge-geodetic sets in graphs

TL;DR

This paper studies monitoring edge-geodetic sets (MEG-sets) and the MEG-number $meg(G)$, formalizing a network monitoring model where a subset of probes monitors every edge via shortest-path constraints. It relates $meg(G)$ to established geodetic-type parameters, proving the chain $g(G)\le eg(G)\le seg(G)\le meg(G)$ and linking MEGs to distance-edge monitoring sets through $dem(G)$. It provides a general upper bound $meg(G)\le \frac{4|V(G)|}{g-3}$ for $2$-connected graphs with girth $g\ge4$, with chromatic-number refinements and corollaries for high-girth graphs, and characterizes graphs with $meg(G)=|V(G)|$ via a vertex-condition, further exploring MEG-extremal families. The work also analyzes how MEG-sets behave under clique-sums and subdivisions, establishing near-tight bounds and showing that certain graph products preserve MEG-extremality, while others may fail; it includes constructive examples realizing prescribed parameter values across multiple families. Together, these results advance the structural theory of MEG-sets and provide tools for designing and analyzing network-monitoring schemes on diverse graph classes.

Abstract

A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs $G$ that have $V(G)$ as their minimum MEG-set, which settles an open problem due to Foucaud \textit{et al.} (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for $meg(G)$ for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of $G$. We examine the change in $meg(G)$ with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.

Figures (5)

• Figure 1: Relations between the parameter $meg$ and other structural parameters in graphs (with no isolated vertices). For the relationships of distance edge-monitoring sets, see foucaud2022monitoring. Edges between parameters indicate that the value of the bottom parameter is upper-bounded by a function of the top parameter. Figure reproduced from foucaud2023monitoringfull.
• Figure 2: A graph $G$ with $2 = g(G) < 3 = eg(G) < 4= seg(G) < 5 = meg(G).$ Note that, a minimum geodetic set of $G$ is $\{v_3,v_5\}$, a minimum edge-geodetic set of $G$ is $\{v_1, v_2, v_4\}$, a minimum strong edge-geodetic set of $G$ is $\{v_1,v_2,v_3,v_4\}$ (the assigned shortest paths between a pair of adjacent vertices is the edge between them, and between a pair of non-adjacent vertices is the $2$-path through $v_5$) and a minimum MEG-set of $G$ is $\{v_1,v_2,v_3,v_4,v_5\}$.
• Figure 3: The structure of $G_{a,b,c,d}.$ The red dashed edge between $z_2$ and $w_1$ represents an edge/non-edge depending on the values of $a$ and $b$.
• Figure 4: A $2$-connected interval graph with a non-extremal MEG-set. Note that the set of all vertices except the universal vertex is an MEG-set.
• Figure 5: The tensor product of a $K_2$ graph $uv$ and the $P_3$ graph $abc$. Here, the set of degree 1 vertices is an MEG-set.

Theorems & Definitions (54)

• Definition 1.1
• Lemma 2.1: Lemma 2.1 of foucaud2023monitoring
• Lemma 2.2: Lemma 2.2 of foucaud2023monitoring
• Proposition 3.1
• proof
• Remark 3.2
• Theorem 3.3
• proof
• Lemma 3.4
• proof