On dominating sets of maximal outerplanar and planar graphs

Li, Zepeng; Zhu, Enqiang; Shao, Zehui; Xu, Jin

doi:10.1016/j.dam.2015.06.024

Cited by 29 publications

(21 citation statements)

References 14 publications

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“…In this paper, we first obtain a bound for the domination number of a maximal outerplane graph based on essential pairs, which corrects the result of Li et al [6], as follows.…”

Section: Introductionsupporting

confidence: 77%

“…Another main result (Theorem 3.2) of Li et al [6] claims that every Hamiltonian plane triangulation with at least seven vertices has domination number at least 5n/16, which is better than the result obtained in [8].…”

Section: Introductionmentioning

confidence: 87%

“…(For example, the pair of white vertices in Figure 2 is an essential pair.) The first vertex in an essential pair is called a bad vertex in [6]. Clearly, the number of essential pairs of G is equal to the number of bad vertices of G. In 2016, Li, et al [6] realized that the degree 2 vertices which require more vertices to dominate are precisely the bad vertices.…”

Section: Introductionmentioning

confidence: 99%

“…The first vertex in an essential pair is called a bad vertex in [6]. Clearly, the number of essential pairs of G is equal to the number of bad vertices of G. In 2016, Li, et al [6] realized that the degree 2 vertices which require more vertices to dominate are precisely the bad vertices. They claimed that the domination number for an n-vertex maximal outerplane graph with k bad vertices (or essential pairs) is at most (n + k)/4.…”

Section: Introductionmentioning

confidence: 99%

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Dominating maximal outerplane graphs and Hamiltonian plane triangulations

Plummer

Zha

2020

Discrete Applied Mathematics

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Let G be a graph and γ(G) denote the domination number of G, i.e. the cardinality of a smallest set of vertices S such that every vertex of G is either in S or adjacent to a vertex in S. Matheson and Tarjan conjectured that a plane triangulation with a sufficiently large number n of vertices has γ(G) ≤ n/4. Their conjecture remains unsettled. In the present paper, we show that: (1) a maximal outerplane graph with n vertices has γ(G) ≤ ⌈ n+k 4 ⌉ where k is the number of pairs of consecutive degree 2 vertices separated by distance at least 3 on the boundary of G; and (2) a Hamiltonian plane triangulation G with n ≥ 23 vertices has γ(G) ≤ 5n/16. We also point out and provide counterexamples for several recent published results of Li et al in [Discrete Appl. Math.198 (2016) 164-169] on this topic which are incorrect.

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“…In this paper, we first obtain a bound for the domination number of a maximal outerplane graph based on essential pairs, which corrects the result of Li et al [6], as follows.…”

Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dominating maximal outerplane graphs and Hamiltonian plane triangulations

Plummer

Zha

2020

Discrete Applied Mathematics

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“…In 2016, Zepeng Li, Enqiang Zhu, Zehui Shao, and Jin Xu, proved the following results on maximal outer planar graph by using edge contraction [42].…”

Section: Theorem[41]mentioning

confidence: 99%

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Yamuna¹,

Karthika²

2016

IJET

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Abstract-A dominating set of G is a set of vertices of G such that every vertex in V -D is adjacent to a vertex in D.The domination number of G, denoted by  ( G ), is the minimum cardinality of a dominating set. Different kinds of graph operations have different outcomes on the domination number of a graph. In this paper we present a brief survey on the impact of different kinds of graph operation on the domination number of a graph.Keyword -Domination, Domination Number, Graph Operation. I. INTRODUCTIONThe study of domination number in graph theory was introduced by Claude Berge in 1958, on his book "Theory of Graphs and its Applications" [1]. But domination number was studied in the name of coefficient of external stability. At the first time the name dominating set and domination number was used by Oyestein Ore in [2]. At the beginning stage the domination number was denoted by d ( G ). Later, in 1977, the notation  ( G ) is used to denote the domination number in Since the evolution of domination theory, types of domination are also established and studied by many researchers. But the original dominating set is the base for all types of domination. Different kinds of graph operations change the graphs and hence influence the domination number of the graph also. In this survey we concentrate on the effect of various kinds of graph operations on the domination number. There are several results relating graph operations and different types of domination. But we restrict our survey only to graph operations and dominating sets. Several results are available in this regard, and many have been omitted in this brief survey. We apologize to the authors for the omission. Many results omitted here can be found in [4][5] and in the articles in the reference section. This survey is restricted to presentation of possible results that describe the effects of the graph operations on the domination number of a graph. The connectivity k = k ( G ) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. A set of independent edges in a graph G is called a matching of G. A 1 -factor or a perfect matching of a graph is a partition of its vertices into adjacent pairs. II. GRAPH THEORY TERMINOLOGY AND CONCEPTSP n , C n , K n , denotes the path, cycle and complete graph with n vertices respectively. The wheel graph is denoted by W n , and it is defined as K 1 + C n -1 . We denote the radius and diameter of G by rad( G ) and diam ( G ), respectively. The minimum and maximum degree on the vertices of G is denoted by  ( G ) and  ( G ) respectively. If  ( G ) =  ( G ) = r, then all points have the same degree and G is called regular of degree r. The regular graphs are those of degree 3, such graphs are called cubic. The complement G of a graph G also has V ( G ) as its vertex set, but two vertices are adjacent in G if and only if they are not adjacent in G.

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Isolation of Regular Graphs and k-Chromatic Graphs

Borg

2024

Mediterr. J. Math.

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Given a set $${\mathcal {F}}$$ F of graphs, we call a copy of a graph in $${\mathcal {F}}$$ F an $${\mathcal {F}}$$ F -graph. The $${\mathcal {F}}$$ F -isolation number of a graph G, denoted by $$\iota (G,{\mathcal {F}})$$ ι ( G , F ) , is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the $${\mathcal {F}}$$ F -graphs contained by G (equivalently, $$G - N[D]$$ G - N [ D ] contains no $${\mathcal {F}}$$ F -graph). Thus, $$\iota (G,\{K_1\})$$ ι ( G , { K 1 } ) is the domination number of G. For any integer $$k \ge 1$$ k ≥ 1 , let $${\mathcal {F}}_{1,k}$$ F 1 , k be the set of regular graphs of degree at least $$k-1$$ k - 1 , let $${\mathcal {F}}_{2,k}$$ F 2 , k be the set of graphs whose chromatic number is at least k, and let $${\mathcal {F}}_{3,k}$$ F 3 , k be the union of $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . Thus, k-cliques are members of both $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . We prove that for each $$i \in \{1, 2, 3\}$$ i ∈ { 1 , 2 , 3 } , $$\frac{m+1}{{k \atopwithdelims ()2} + 2}$$ m + 1 k 2 + 2 is a best possible upper bound on $$\iota (G, {\mathcal {F}}_{i,k})$$ ι ( G , F i , k ) for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.

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On dominating sets of maximal outerplanar and planar graphs

Cited by 29 publications

References 14 publications

Dominating maximal outerplane graphs and Hamiltonian plane triangulations

Dominating maximal outerplane graphs and Hamiltonian plane triangulations

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Isolation of Regular Graphs and k-Chromatic Graphs

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