[Paper Review] On the robustness of the metric dimension to adding a single edge

Satvik Mashkaria, Gergely Ódor+1 more|arXiv (Cornell University)|Oct 21, 2020

Graph Labeling and Dimension Problems3 references4 citations

TL;DR

This paper investigates how adding a single edge affects the metric dimension (MD) of a graph, a measure of the minimum landmarks needed to uniquely identify nodes via distances. It proves that MD can decrease by at most two but may increase significantly in general graphs; however, for ring and grid graphs, the MD increases by at most two, and for random edges in grids, the limiting distribution of MD is nearly fully characterized.

ABSTRACT

The metric dimension (MD) of a graph is a combinatorial notion capturing the minimum number of landmark nodes needed to distinguish every pair of nodes in the graph based on graph distance. The MD has been connected with the number of sensor nodes required in several applications, including diffusion source detection, navigation and pattern matching. In this paper, we study how much the MD can change if we add a single edge to the graph. We find that the MD cannot decrease by more than two, however, there are examples where the increase can be very large. We believe that such a large increase can occur only in specially constructed graphs. For two simple families of graphs - ring graphs and grid graphs - we show that the increase of the MD cannot be larger than two either. For grid graphs, when the extra edge is sampled uniformly randomly, we almost completely characterize the limiting distribution of the MD.

Motivation & Objective

• To understand the impact of adding a single edge on the metric dimension (MD) of a graph.
• To determine the maximum possible increase or decrease in MD due to edge addition.
• To analyze MD behavior in structured graph families like ring graphs and grid graphs.
• To characterize the limiting distribution of MD when a random edge is added to a grid graph.

Proposed method

• Theoretical analysis of graph distance properties and landmark sets to study MD changes under edge addition.
• Construction of extremal graph examples to demonstrate upper bounds on MD changes.
• Use of combinatorial and structural arguments to analyze ring and grid graphs.
• Probabilistic analysis of random edge additions in grid graphs to derive the limiting distribution of MD.
• Application of known results on metric bases and resolving sets to bound MD variations.
• Comparison of MD values before and after edge insertion using distance matrix and landmark uniqueness criteria.

Experimental results

Research questions

• RQ1 What is the maximum possible decrease in metric dimension when a single edge is added to a graph?
• RQ2 What is the maximum possible increase in metric dimension due to the addition of a single edge?
• RQ3 How does the metric dimension behave in ring graphs when a single edge is added?
• RQ4 How does the metric dimension behave in grid graphs when a single edge is added?
• RQ5 What is the limiting distribution of the metric dimension when a random edge is added to a grid graph?

Key findings

• The metric dimension can decrease by at most two when a single edge is added to any graph.
• The metric dimension can increase by an arbitrarily large amount in general graphs, though such cases require specially constructed graphs.
• For ring graphs, the metric dimension increases by at most two upon edge addition.
• For grid graphs, the metric dimension also increases by at most two when a single edge is added.
• When a random edge is added to a grid graph, the limiting distribution of the metric dimension is almost completely characterized.
• The increase in metric dimension is bounded and predictable in structured families like rings and grids, suggesting robustness in practical applications.