This question explores the reasoning behind the exclusion of edges when defining or analyzing resolvability parameters in graph theory. Resolvability parameters, such as the metric dimension, edge metric dimension, or mixed metric dimension, typically involve selecting subsets of vertices or combinations of vertices and edges to uniquely identify other elements of the graph based on distances or other structural properties. The question challenges the conventional approach, asking why edges alone are not included as part of the resolving set in these definitions
I found a couple of instances of articles that use edges to resolve other edges, namely:
Edge metric dimension of graphs, Nasir, Zafar, Zahid
Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph, Liu, Zahid, Nasir, Nazeer.
Of course this is really equivalent to the ordinary metric dimension in a line graph, see
On the metric dimension of line graphs, Feng, Xu, Wang.
I haven't found any articles yet that resolve vertices by edges. But of course such a set doesn't always exist, e.g. K_2.
dear James Tuite no. I think you did not read those articles I copied these lins from your reminded article. They take R from the vertices set. For any two vertices 𝑥,𝑦x,y∈𝑉𝐺,VG, the distance 𝑑(𝑥,𝑦)d(x,y) is the length of a shortest path between x and y. Let 𝑅={𝑟1,𝑟2,…,𝑟𝑙}⊂𝑉𝐺R={r1,r2,…,rl}⊂VG be an ordered set and let x∈𝑉𝐺,VG, then 𝑟(𝑥,𝑅)r(x,R) representation of x with respect to R is the l-tuple (𝑑(𝑥,𝑟1),𝑑(𝑥,𝑟2),…,𝑑(𝑥,𝑟𝑙))d(x,r1),d(x,r2),…,d(x,rl).
Hello. Both of the first two articles resolve edges with an edge subset. Here are the links. See definitions 1.2 and 1 respectively.
https://www.researchgate.net/publication/322634658_Edge_metric_dimension_of_graphs
https://www.mdpi.com/2227-7390/6/11/243
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