Domination and Outer Connected Domination in Maximal Outerplanar Graphs

Zhuang, Wei

doi:10.1007/s00373-021-02383-w

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…Various types of domination of maximal outerplanar graphs were widely studied in literature, such as secure domination [3], partial domination [5], domination [6,12,14], total domination [8,9], semipaired domination [10] and connected domination [18]. Very recently, Zhuang [17] proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1 [17]. If G is a maximal outerplanar graph of order n ≥ 3, and k is the number of vertices of degree 2 in G, then γc (G)…”

Section: Introductionmentioning

confidence: 99%

“…Thus, γc (G u ) ≤ γc (G) − 1. Now, we are ready to give the construction of a family graphs A which was construct by Zhuang in [17]. Each graph in Figures 4 and 5 is called "basic graph", and each graph depicted in Figure 6 is called "gadget".…”

mentioning

confidence: 99%

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Outer connected domination in maximal outerplanar graphs and beyond

Yang¹,

Wu²

2024

Discuss. Math. Graph Theory

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A set S of vertices in a graph G is an outer connected dominating set of G if every vertex in V \S is adjacent to a vertex in S and the subgraph induced by V \S is connected. The outer connected domination number of G, denoted by γc (G), is the minimum cardinality of an outer connected dominating set of G. Zhuang [Domination and outer connected domination in maximal outerplanar graphs, Graphs Combin. 37 (2021) 2679-2696] recently proved that γc (G) ≤ n+k 4 for any maximal outerplanar graph G of order n ≥ 3 with k vertices of degree 2 and posed a conjecture which states that G is a striped maximal outerplanar graph with γc (G) = n+2 4 if and only if G ∈ A, where A consists of six special families of striped outerplanar graphs. We disprove the conjecture. Moreover, we show that the conjecture become valid under some additional property to the striped maximal outerplanar graphs. In addition, we extend the above theorem of Zhuang to all maximal K 2,3 -minor free graphs without K 4 and all K 4 -minor free graphs.

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Section: Introductionmentioning

confidence: 99%

“…Theorem 1 [17]. If G is a maximal outerplanar graph of order n ≥ 3, and k is the number of vertices of degree 2 in G, then γc (G)…”

Section: Introductionmentioning

confidence: 99%

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Outer connected domination in maximal outerplanar graphs and beyond

Yang¹,

Wu²

2024

Discuss. Math. Graph Theory

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“…In the same paper, the authors conjectured that γ(G) ≤ n 4 for every n-vertex triangulation G, when n is large enough. Since then, several papers have been devoted to either trying to prove that conjecture [23,31,34], or showing thigh bounds, mainly for MOPs, for several variants of the domination number [2,3,4,6,7,9,10,19,26,31,35,37]. In particular, Canales et al [3] proved that γ pr (G) ≤ 2 n 4 for any MOP G of order n ≥ 4, and Henning and Kaemawichanurat [19] showed that γ pr2 (G) ≤ 2 5 n for any MOP G of order n ≥ 5, except for a special family F of MOPs of order 9.…”

Section: Introductionmentioning

confidence: 99%

Paired and semipaired domination in triangulations

Claverol¹,

Hernando²,

Maureso³

et al. 2022

Preprint

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A dominating set of a graph G is a subset D of vertices such that every vertex not in D is adjacent to at least one vertex in D. A dominating set D is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in D is paired with exactly one other vertex in D that is within distance 2 from it. The paired domination number, denoted by γ pr (G), is the minimum cardinality of a paired dominating set of G, and the semipaired domination number, denoted by γ pr2 (G), is the minimum cardinality of a semipaired dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ pr (G) ≤ 2 n 4 for any neartriangulation G of order n ≥ 4, and that with some exceptions, γ pr2 (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5.

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The Extended Dominating Sets in Graphs

Gao

Shi

et al. 2023

Asia Pac. J. Oper. Res.

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Let [Formula: see text] be a graph and let [Formula: see text] be an integer. A vertex subset [Formula: see text] is called a [Formula: see text]-extended dominating set if every vertex [Formula: see text] of [Formula: see text] satisfies one of the following conditions: the distance between [Formula: see text] and [Formula: see text] is at most one or there are at least [Formula: see text] different vertices [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] ([Formula: see text]) is two. The [Formula: see text]-extended domination number [Formula: see text] of [Formula: see text] is the minimum size over all [Formula: see text]-extended dominating sets in [Formula: see text]. When [Formula: see text], they are called the extended dominating set and the extended domination number of [Formula: see text], respectively. In this paper, we mainly study the bounds of the extended domination numbers of graphs. First, we obtain the exact values of the extended domination numbers for paths and cycles. And then the Nordhaus–Gaddum bounds for the extended domination numbers are provided. Additionally, we give some bounds of the extended domination numbers for planar graphs with small diameters. Finally, we consider the [Formula: see text]-extended domination numbers in Random graphs.

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Domination and Outer Connected Domination in Maximal Outerplanar Graphs

Cited by 3 publications

References 15 publications

Outer connected domination in maximal outerplanar graphs and beyond

Outer connected domination in maximal outerplanar graphs and beyond

Paired and semipaired domination in triangulations

The Extended Dominating Sets in Graphs

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