Volume 56, Number 1, January-February 2022
|Page(s)||199 - 211|
|Published online||07 February 2022|
Incidence dimension and 2-packing number in graphs
Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, 2000 Maribor, Slovenia
2 Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, 1000 Ljubljana, Slovenia
3 Departamento de Matemáticas, Escuela Técnica Superior de Ingeniería de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, 11202 Algeciras, Spain
* Corresponding author: firstname.lastname@example.org
Accepted: 3 January 2020
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 ∈ 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 dimI(G). A set of vertices P ⊆ V(G) is a 2-packing of G if the distance in G between any pair of distinct vertices from P is larger than two. The largest cardinality of a 2-packing of G is the packing number of G and is denoted by ρ(G). In this article, the incidence dimension is introduced and studied. The given results show a close relationship between dimI(G) and ρ(G). We first note that the complement of any 2-packing in graph G is an incidence generator for G, and further show that either dimI(G) = |V(G)-|ρ(G) or dimI(G) = |V(G)-|ρ(G) - 1 for any graph G. In addition, we present some bounds for dimI(G) and prove that the problem of determining the incidence dimension of a graph is NP-hard.
Mathematics Subject Classification: 05C69 / 05C12
Key words: incidence dimension / incidence generator / 2-packing
© The authors. Published by EDP Sciences, ROADEF, SMAI 2022
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