Domination in digraphs and their products

Abstract

A dominating (respectively, total dominating) set S of a digraph D is a set of vertices in D such that the union of the closed (respectively, open) out-neighborhoods of vertices in S equals the vertex set of D. The minimum size of a dominating (respectively, total dominating) set of D is the domination (respectively, total domination) number of D, denoted γ(D) (respectively, γ t (D)). The maximum number of pairwise disjoint closed (respectively, open) in-neighborhoods of D is denoted by ρ(D) (respectively, ρ o (D)). We prove that in digraphs whose underlying graphs have girth at least 7, the closed (respectively, open) in-neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree T (that is, a digraph whose underlying graph is a tree), γ t (T ) = ρ o (T ) and γ(T ) = ρ(T ). By using the former equality we then prove that γ t (G × T ) = γ t (G)γ t (T ), where G is any digraph and T is any ditree, each without a source vertex, and G × T is their direct product. From the equality γ(T ) = ρ(T ) we derive the bound γ(G T ) ≥ γ(G)γ(T ), where G is an arbitrary digraph, T an arbitrary ditree and G T is their Cartesian product. In general digraphs this Vizing-type bound fails, yet we prove that for any digraphs G and H, where γ(G) ≥ γ(H), we have γ(G H) ≥ 1 2 γ(G)(γ(H) + 1). This inequality is sharp as demonstrated by an infinite family of examples. Ditrees T and digraphs H enjoying γ(T H) = γ(T )γ(H) are also investigated.

Coloring of two-step graphs: open packing partitioning of graphs

“…For a given graph G, the closed neighborhood graph of G, denoted by N c (G), has the vertex set V (G); two distinct vertices u and v are adjacent in N c (G) if and only if N G [u]∩N G [v] = ∅ (see [3]). In fact, a 2-distance coloring of a graph G is the same as coloring of its closed neighborhood graph.…”