Dominating maximal outerplane graphs and Hamiltonian plane triangulations

Plummer, Michael D.; Ye, Dong; Zha, Xiaoya

doi:10.1016/j.dam.2019.12.003

Cited by 6 publications

(2 citation statements)

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“…Conjecture 1 is best possible since, for any planar triangulation G that consists of a triangulation of k vertex-disjoint copies of K 4 embedded so that each copy of K 4 is in the outer face of all other copies of K 4 , we have γ(G) = k = n/4. In spite of the efforts of specialists [7,8,10,11], the best general upper bound found so far for γ(G) is due to Špacapan [13], who proved that γ(G) 17n/53 for every planar triangulation G on n 6 vertices. Observe also that the bound of Conjecture 1 is not satisfied for near triangulations [9, Figure 1].…”

Section: Introductionmentioning

confidence: 99%

Independent Dominating Sets in Planar Triangulations

Botler,

Fernandes,

Gutiérrez

2024

Electron. J. Combin.

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In 1996, Matheson and Tarjan proved that every near planar triangulation on $n$ vertices contains a dominating set of size at most $n/3$, and conjectured that this upper bound can be reduced to $n/4$ for planar triangulations when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum $\varepsilon$ for which every near planar triangulation on $n$ vertices contains an independent dominating set of size at most $\varepsilon n$? We prove that $2/7 \leq \varepsilon \leq 5/12$. Moreover, this upper bound can be improved to $3/8$ for planar triangulations, and to $1/3$ for planar triangulations with minimum degree 5.

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Section: Introductionmentioning

confidence: 99%

Independent Dominating Sets in Planar Triangulations

Botler,

Fernandes,

Gutiérrez

2024

Electron. J. Combin.

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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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