Incidence dimension and 2-packing number in graphs

Let \(G=(V,E)\) be a graph. A set of vertices \(A\) is an incidence generator for \(G\) if for any two distinct edges \(e,f\in E(G)\) there exists a vertex from \(A\) which is an endpoint of either \(e\) or \(f\). The smallest cardinality of an incidence generator for \(G\) is called the incidence dimension and is denoted by \(dim_I(G)\). A set of vertices \(P\) is a 2-packing if the distance between any pair of distinct vertices from \(P\) is greater than two. The largest cardinality of a 2-packing of \(G\) is the packing number of \(G\) and is denoted by \(\rho(G)\). The incidence dimension of graphs is introduced and studied in this article, and we emphasize in the closed relationship between \(dim_I(G)\) and \(\rho(G)\). We first note that the complement of any 2-packing in a graph \(G\) is always an incidence generator for \(G\), and further show that either \(dim_I(G)=\rho(G)\) or \(dim_I(G)=\rho(G)-1\) for any graph \(G\). In addition, we also prove that the problem of determining the incidence dimension of a graph is NP-complete, and present some bounds for it.