AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286

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On the edge metric dimension of graphs

1 College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
2 School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, China
3 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, Zhejiang, China

Let $G=(V,E)$ be a connected graph of order $n$. $S \subseteq V$ is an edge metric generator of $G$ if any pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of a graph $G$ is the least size of an edge metric generator of $G$. In this paper, we give the characterization of all connected bipartite graphs with $edim=n-2$, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for $edim(G)=n-2$, where $G$ is a graph with maximum degree $n-1$. In addition, the relationship between the edge metric dimension and the clique number of a graph $G$ is investigated by construction.
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