Abstract
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.
Highlights
The famous Gallai’s theorem states that α(G) + β(G) = n, where G is a graph on n vertices, β(G) is the vertex covering number of G and α(G) is the independence number of G
One possibility arises from observing an independent set from a different perspective
As a kind of a mix with respect to the last two parameters, we introduce the concept of incidence dimension in graphs which arises from the two concepts above in some natural way of research evolution
Summary
Any vertex cover set S of cardinality β(G) determines an independent set I of cardinality α(G), which is the complement of S. With such elegance, the hunt for analog results is always open. One possibility arises from observing an independent set from a different (but equivalent) perspective. It comes from the notion of k-packings in graphs.
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