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Upper distance-two domination

Hedetniemi Jason T., Hedetniemi Stephen T., Lewis Thomas M.

Opuscula Mathematica

2025

Vol. 45, no. 3

339--350

Let G = (V,E) be a graph with vertex set V and edge set E. A set S ⊂ V is a 2-packing in G if for any two vertices u, v ∈ S, the distance between them satisfies d(u, v) > 2. The upper 2-packing number P2(G) is the maximum cardinality of a 2-packing in G. A set S ⊂ V is a dominating set for G if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. A set S ⊂ V is a distance-2 dominating set if for every vertex v ∈ V − S there exists a vertex u ∈ S such that d(u, v) ≤ 2. The upper distance-2 domination number Γ≤2(G) is the maximum cardinality of a minimal distance-2 dominating set in G. In this paper we establish two families of graphs G for which P2(G) = γ(G) = Γ≤2(G), which extend several well-known equalities of the form P2(G) = γ(G).