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 ( ⌊ 2 D 3 ⌋ + 1 ) k + k ∑ i = 1 ⌈ D 3 ⌉ ( 2 i ) k − 1 , sharpening the bound of k 2 + k D k − 1 + D k from Zubrilina (2018). We also show that the maximum value of n for which some graph of metric dimension ≤ k contains the complete graph K n as a subgraph is n = 2 k . We prove that the maximum value of n for which some graph of metric dimension ≤ k contains the complete bipartite graph K n , n as a subgraph is 2 Θ ( k ) . Furthermore, we show that the maximum value of n for which some graph of edge metric dimension ≤ k contains K 1 , n as a subgraph is n = 2 k . We also show that the maximum value of n for which some graph of metric dimension ≤ k contains K 1 , n as a subgraph is 3 k − O ( k ) . In addition, we prove that the d -dimensional grids ∏ i = 1 d P r i 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 3 k − 1 .