Metric dimension and pattern avoidance in graphs

Abstract

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter D and edge metric dimension k is at most (⌊2D3⌋+1)k+k∑i=1⌈D3⌉(2i)k−1, sharpening the bound of k2+kDk−1+Dk from Zubrilina (2018). We also show that the maximum value of n for which some graph of metric dimension ≤k contains the complete graph Kn as a subgraph is n=2k. We prove that the maximum value of n for which some graph of metric dimension ≤k contains the complete bipartite graph Kn,n as a subgraph is 2Θ(k). Furthermore, we show that the maximum value of n for which some graph of edge metric dimension ≤k contains K1,n as a subgraph is n=2k. We also show that the maximum value of n for which some graph of metric dimension ≤k contains K1,n as a subgraph is 3k−O(k).In addition, we prove that the d-dimensional grids ∏i=1dPri have edge metric dimension at most d. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension 2 and that d-dimensional hypercubes have edge metric dimension at most d. We also provide a characterization of n-vertex graphs with edge metric dimension n−2, answering a question of Zubrilina. As a result of this characterization, we prove that any connected n-vertex graph G such that edim(G)=n−2 has diameter at most 5. More generally, we prove that any connected n-vertex graph with edge metric dimension n−k has diameter at most 3k−1.