Paper Info
Title
On the robustness of the metric dimension of grid graphs to adding a single 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 + Ber(8/27), and we give an almost complete proof. (C) 2022 The Author(s). Published by Elsevier B.V.
Year
DOI
Venue
2022
10.1016/j.dam.2022.02.014
DISCRETE APPLIED MATHEMATICS
Keywords
DocType
Volume
Metric dimension, Extra edge
Journal
316
ISSN
Citations 
PageRank 
0166-218X
0
0.34
References 
Authors
0
3
Name
Order
Citations
PageRank
Satvik Mashkaria100.34
Odor Gergely201.35
Patrick Thiran32712217.24