Large planar graphs with given diameter and maximum degree

Fellows, Michael R.; Hell, Pavol; Seyffarth, Karen

doi:10.1016/0166-218x(94)00011-2

Cited by 34 publications

(18 citation statements)

References 4 publications

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“…With this result, and with a similar approach as in Fellows, Hell, and Seyffarth [2], that is using Theorem 3.1, we obtain the following theorem. (a) If ∆ = 4:…”

Section: ⌊D/2⌋mentioning

confidence: 82%

“…Fellows, Hell and Seyffarth [2] obtained bounds on the (∆, D) problem for planar graphs applying the following theorem by Lipton and Tarjan [4]. Clearly, this theorem also holds for maximal planar bipartite graphs.…”

Section: An Upper Boundmentioning

confidence: 93%

“…The upper bound given by Fellows, Hell and Seyffarth [2] for planar graphs is n ≤ 3(2D + 1)(2∆ ⌊D/2⌋ + 1).…”

Section: ⌊D/2⌋mentioning

confidence: 99%

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The degree/diameter problem in maximal planar bipartite graphs

Dalfó

Huemer

Salas³

2014

Electronic Notes in Discrete Mathematics

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The (∆, D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree ∆ and diameter D. We consider the (∆, D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (∆, 2) problem, the number of vertices is n = ∆+2; and for the (∆, 3) problem, n = 3∆−1 if ∆ is odd and n = 3∆ − 2 if ∆ is even. Then, we study the general case (∆, D) and obtain that an upper bound on n is approximately 3(2D + 1)(∆ − 2) ⌊D/2⌋ and another one is C(∆ − 2) ⌊D/2⌋ if ∆ ≥ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (∆ − 2) k if D = 2k, and 3(∆ − 3) k if D = 2k + 1, for ∆ and D sufficiently large in both cases.

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“…With this result, and with a similar approach as in Fellows, Hell, and Seyffarth [2], that is using Theorem 3.1, we obtain the following theorem. (a) If ∆ = 4:…”

Section: ⌊D/2⌋mentioning

confidence: 82%

Section: An Upper Boundmentioning

confidence: 93%

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The degree/diameter problem in maximal planar bipartite graphs

Dalfó

Huemer

Salas³

2014

Electronic Notes in Discrete Mathematics

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“…So the interesting question is what happens when the diameter is 2 or 3. As pointed out in [7], the restriction of bounding the diameter on the domination of a planar graph is reasonable to impose because planar graphs with small diameter are often important in applications (see [2]). MacGillivray and Seyffarth [7] proved that planar graphs with diameter two or three have bounded domination numbers.…”

Section: Introductionmentioning

confidence: 99%

Domination in planar graphs with small diameter*

Goddard

Henning²

2002

Journal of Graph Theory

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MacGillivray and Seyffarth (J Graph Theory 22 (1996), 213-229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed. ß

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“…An algorithm that controls both the degree distribution and assortativity can be found in Xulvi-Brunet and Sokolov [49]. The problem of constructing graphs with given maximum degree and diameter that has the largest number of vertices has also been studied [3,7,16]. In addition, Nakano et al [30] discuss an algorithm to construct weighted graphs such that the total weight of the edges incident to a node is at least the given weight of the node.…”

Section: Introductionmentioning

confidence: 99%

Designing networks: A mixed‐integer linear optimization approach

et al. 2016

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Designing networks with specified collective properties is useful in a variety of application areas, enabling the study of how given properties affect the behavior of network models, the downscaling of empirical networks to workable sizes, and the analysis of network evolution. Despite the importance of the task, there currently exists a gap in our ability to systematically generate networks that adhere to theoretical guarantees for the given property specifications. In this paper, we propose the use of Mixed-Integer Linear Optimization modeling and solution methodologies to address this Network Generation Problem. We present a number of useful modeling techniques and apply them to mathematically express and constrain network properties in the context of an optimization formulation. We then develop complete formulations for the generation of networks that attain specified levels of connectivity, spread, assortativity and robustness, and we illustrate these via a number of computational case studies.

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Large planar graphs with given diameter and maximum degree

Cited by 34 publications

References 4 publications

The degree/diameter problem in maximal planar bipartite graphs

The degree/diameter problem in maximal planar bipartite graphs

Domination in planar graphs with small diameter*

Designing networks: A mixed‐integer linear optimization approach

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