On the edge metric dimension of convex polytopes and its related graphs

Let $$G=(V, E)$$G=(V,E) be a connected graph. The distance between the edge $$e=uv\in E$$e=uv∈E and the vertex $$x\in V$$x∈V is given by $$d(e, x) = \min \{d(u, x), d(v, x)\}$$d(e,x)=min{d(u,x),d(v,x)}. A subset $$S_{E}$$SE of vertices is called an edge metric generator for G if for every two distinct edges $$e_{1}, e_{2}\in E$$e1,e2∈E, there exists a vertex $$x\in S_{E}$$x∈SE such that $$d(e_{1}, x)\ne d(e_{2}, x)$$d(e1,x)≠d(e2,x). An edge metric generator containing a minimum number of vertices is called an edge metric basis for G and the cardinality of an edge metric basis is called the edge metric dimension denoted by $$\mu _{E}(G)$$μE(G). In this paper, we study the edge metric dimension of some classes of plane graphs. It is shown that the edge metric dimension of convex polytope antiprism $$A_{n}$$An, the web graph $${\mathbb {W}}_{n}$$Wn, and convex polytope $${\mathbb {D}}_{n}$$Dn are bounded, while the prism related graph $$D^{*}_{n}$$Dn∗ has unbounded edge metric dimension.