Efficient closed domination in digraph products

Peterin, Iztok; Yero, Ismael G.

doi:10.1007/s10878-018-0370-y

Cited by 3 publications

(4 citation statements)

References 25 publications

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“…The fibers of the direct product

G \times H

are defined as they are for the Cartesian product, but in the direct product they are independent sets. We remark in passing that a few recent papers considered domination in digraph products concentrating on efficient domination [19], rainbow domination [7] and digraph kernels [14].…”

Section: Definitions and Notationmentioning

confidence: 99%

Domination in digraphs and their direct and Cartesian products

Brešar

Kuenzel

Rall

2021

Journal of Graph Theory

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“…The fibers of the direct product

G \times H

Section: Definitions and Notationmentioning

confidence: 99%

Domination in digraphs and their direct and Cartesian products

Brešar

Kuenzel

Rall

2021

Journal of Graph Theory

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“…The fibers of the direct product G × H are defined as they are for the Cartesian product, but in the direct product they are independent sets. We remark in passing that a few recent papers considered domination in digraph products concentrating on efficient domination [19], rainbow domination [7] and digraph kernels [14].…”

Section: Definitions and Notationmentioning

confidence: 99%

Domination in digraphs and their products

Brešar,

Kuenzel,

Rall

2020

Preprint

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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.

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“…The terminology and basic definitions in this section are summarized from [30] where the authors present the results on the efficient closed domination among standard products of digraphs.…”

Section: Preliminariesmentioning

confidence: 99%

“…As in the case of graphs, there is more literature concerning efficient closed domination digraphs than that of efficient open domination digraphs. Here we mention only [30], a recent work that brings the results on the efficient closed domination among standard products of digraphs.…”

Section: Introductionmentioning

confidence: 99%

Efficient Open Domination in Digraph Products

Božović

Peterin

2020

Mathematics

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A digraph D is an efficient open domination digraph if there exists a subset S of V ( D ) for which the open out-neighborhoods centered in the vertices of S form a partition of V ( D ) . In this work we deal with the efficient open domination digraphs among four standard products of digraphs. We present a method for constructing the efficient open domination Cartesian product of digraphs with one fixed factor. In particular, we characterize those for which the first factor has an underlying graph that is a path, a cycle or a star. We also characterize the efficient open domination strong product of digraphs that have factors whose underlying graphs are uni-cyclic graphs. The full characterizations of the efficient open domination direct and lexicographic product of digraphs are also given.

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Efficient closed domination in digraph products

Cited by 3 publications

References 25 publications

Domination in digraphs and their direct and Cartesian products

Domination in digraphs and their direct and Cartesian products

Domination in digraphs and their products

Efficient Open Domination in Digraph Products

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