On the edge dimension and the fractional edge dimension of graphs

Abstract

Let G be a graph with vertex set V ( G ) and edge set E ( G ) , and let d ( u , w ) denote the length of a u − w geodesic in G . For any vertex v ∈ V ( G ) and any edge e = x y ∈ E ( G ) , let d ( e , v ) = min { d ( x , v ) , d ( y , v ) } . For any distinct edges e 1 , e 2 ∈ E ( G ) , let R { e 1 , e 2 } = { z ∈ V ( G ) : d ( z , e 1 ) ≠ d ( z , e 2 ) } . Kelenc, Tratnik and Yero [Discrete Appl. Math. 251 (2018) 204-220] introduced the notion of an edge resolving set and the edge dimension of a graph: A vertex subset S ⊆ V ( G ) is an edge resolving set of G if | S ∩ R { e 1 , e 2 } | ≥ 1 for any distinct edges e 1 , e 2 ∈ E ( G ) , and the edge dimension , edim ( G ) , of G is the minimum cardinality among all edge resolving sets of G . For a function g defined on V ( G ) and for U ⊆ V ( G ) , let g ( U ) = ∑ s ∈ U g ( s ) . A real-valued function g : V ( G ) → [ 0 , 1 ] is an edge resolving function of G if g ( R { e 1 , e 2 } ) ≥ 1 for any distinct edges e 1 , e 2 ∈ E ( G ) . The fractional edge dimension , edim f ( G ) , of G is min { g ( V ( G ) ) : g is an edge resolving function of G } . Note that edim f ( G ) reduces to edim ( G ) if the codomain of edge resolving functions is restricted to { 0 , 1 } . In this paper, we introduce and study the fractional edge dimension of graphs, and we obtain some general results on the edge dimension of graphs. We show that there exist two non-isomorphic graphs on the same vertex set with the same edge metric coordinates. We construct two graphs G and H such that H ⊂ G and both edim ( H ) − edim ( G ) and edim f ( H ) − edim f ( G ) can be arbitrarily large. We show that a graph G with edim ( G ) = 2 cannot have K 5 or K 3 , 3 as a subgraph, and we construct a non-planar graph H satisfying edim ( H ) = 2 . It is easy to see that, for any connected graph G of order at least three, 1 ≤ edim f ( G ) ≤ | V ( G ) | 2 ; we characterize graphs G satisfying edim f ( G ) = 1 and examine some graph classes satisfying edim f ( G ) = | V ( G ) | 2 . We also determine the fractional edge dimension for some classes of graphs.