Total domination in maximal outerplanar graphs

Dorfling, Michael; Hattingh, Johannes H.; Jonck, Elizabeth

doi:10.1016/j.dam.2016.10.020

Cited by 23 publications

(11 citation statements)

References 6 publications

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“…Thus, we improve the best known upper bound 6 11 n on the total domination number of n-vertex near-triangulations. This last bound follows from the fact that a near-triangulation is 2-connected and from the following result proved in [13]: If G is a 2-connected graph of order n > 18, then γ t (G) ≤ 6 11 n. The upper bound 2n…”

Section: Introductionmentioning

confidence: 95%

“…In this paper, we extend the result proved in [5,17] to the family of neartriangulations and we show that γ t (G) ≤ 2n for any near-triangulation G of order n ≥ 5, apart from the graphs H 1 and H 2 . Thus, we improve the best known upper bound 6 11 n on the total domination number of n-vertex near-triangulations. This last bound follows from the fact that a near-triangulation is 2-connected and from the following result proved in [13]: If G is a 2-connected graph of order n > 18, then γ t (G) ≤ 6 11 n. The upper bound 2n 5 on the total domination number in near-triangulations is proved in Section 4.…”

Section: Introductionmentioning

confidence: 95%

“…In fact, the upper bound n 3 on the domination number in MOPs was already implicitly proved by Fisk [8]. In [1,23], it is shown that γ(G) ≤ (n + k)/4, where k is the number of vertices of degree 2 in a MOP G. Dorfling et al proved in [6] that γ t (G) ≤ n+k 3 for a MOP G of order n with k vertices of degree 2. The same authors proved in [5] that apart from the graphs H 1 and H 2 shown in Figure 1, γ t (G) ≤ 2n 5 for a MOP G of order n ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

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Total domination in plane triangulations

Aguas

García

Peñalver

et al. 2021

Discrete Mathematics

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A total dominating set of a graph G = (V, E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total 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 t (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5, with two exceptions.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Total domination in plane triangulations

Aguas

García

Peñalver

et al. 2021

Discrete Mathematics

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“…Li et al [15] improved the result by showing that γ(G) ≤ n+t 4 , where t is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. For results on other types of domination in maximal outerplanar graphs, we refer the reader to [1,3,7,8,12,14,19,20].…”

Section: Introductionmentioning

confidence: 99%

Connected domination in maximal outerplanar graphs

Zhuang

2020

Discrete Applied Mathematics

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“…Fisk [8] and Matheson and Tarjan [12] also gave alternative proofs. For results on other types of domination in mops, we refer the reader to [3,5,6,7,9,15]. Caro and Hansberg [2] proved that the K 1,1isolation number of a mop of order n ≥ 4 is at most n/4.…”

Section: Introductionmentioning

confidence: 99%

A generalization of the Art Gallery Theorem

Borg¹,

Kaemawichanurat²

2020

Preprint

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Several domination results have been obtained for maximal outerplanar graphs (mops).The classical domination problem is to minimize the size of a set S of vertices of an n-vertex graph G such that G − N [S], the graph obtained by deleting the closed neighborhood of S, contains no vertices. A classical result of Chvátal, the Art Gallery Theorem, tells us that the minimum size is at most n/3 if G is a mop. Here we consider a modification by allowing G − N [S] to have a maximum degree of at most k. Let ι k (G) denote the size of a smallest set S for which this is achieved. If n ≤ 2k + 3, then trivially ι k (G) ≤ 1. Let G be a mop on n ≥ max{5, 2k + 3} vertices, n 2 of which are of degree 2. Sharp bounds on ι k (G) have been obtained for k = 0 and k = 1, namely ι 0 (G) ≤ min{ n 4 , n+n2 5 , n−n2 3 } and ι 1 (G) ≤ min{ n 5 , n+n2 6 , n−n2 3 }. We prove that ι k (G) ≤ min{ n k+4 , n+n2 k+5 , n−n2 k+2 } for any k ≥ 0, and that this bound is sharp. We also prove that n−n2 2 is a sharp upper bound on the domination number of G.

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Total domination in maximal outerplanar graphs

Cited by 23 publications

References 6 publications

Total domination in plane triangulations

Total domination in plane triangulations

Connected domination in maximal outerplanar graphs

A generalization of the Art Gallery Theorem

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