On the robustness of the metric dimension of grid graphs to adding a single edge

TL;DR

This paper investigates how adding a single edge affects the metric dimension of grid graphs, showing it is usually at most doubled, with potential for exponential increase in general graphs, and provides bounds and conjectures for grid graphs.

Contribution

It demonstrates that in grid graphs, the metric dimension increases at most by a factor of two when adding one edge, contrasting with possible exponential increases in general graphs.

Findings

In grid graphs, the metric dimension is at most doubled by adding one edge.

Constructed examples show exponential increases possible in general graphs.

Conjecture and partial proof that MD converges to a specific distribution when adding a random edge.

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. We study how much the MD can increase if we add a single edge to the graph. The extra edge can either be selected adversarially, in which case we are interested in the largest possible value that the MD can take, or uniformly at random, in which case we are interested in the distribution of the MD. The adversarial setting has already been studied by [Eroh et. al., 2015] for general graphs, who found an example where the MD doubles on adding a single edge. By constructing a different example, we show that this increase can be as large as exponential. However, we believe that such a large increase can occur only in specially constructed graphs, and that in most interesting graph families, the MD at most doubles on adding a single edge. We prove this for $d$-dimensional grid graphs, by showing that $2d$ appropriately chosen corners and the endpoints of the extra edge can distinguish every pair of nodes, no matter where the edge is added. For the special case of $d=2$, we show that it suffices to choose the four corners as landmarks. Finally, when the extra edge is sampled uniformly at random, we conjecture that the MD of 2-dimensional grids converges in probability to $3+\mathrm{Ber}(8/27)$, and we give an almost complete proof.