TY - JOUR

T1 - On the robustness of the metric dimension of grid graphs to adding a single edge

AU - Mashkaria, Satvik

AU - Ódor, Gergely

AU - Thiran, Patrick

N1 - Publisher Copyright:
© 2022 The Author(s)

PY - 2022/7/31

Y1 - 2022/7/31

N2 - 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+Ber(8/27), and we give an almost complete proof.

AB - 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+Ber(8/27), and we give an almost complete proof.

KW - Extra edge

KW - Metric dimension

UR - http://www.scopus.com/inward/record.url?scp=85128134588&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2022.02.014

DO - 10.1016/j.dam.2022.02.014

M3 - Article

AN - SCOPUS:85128134588

SN - 0166-218X

VL - 316

SP - 1

EP - 27

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -