Degree diameter problem on honeycomb networks

Holub, Přemysl; Miller, Mirka; Pérez-Rosés, Hebert; Ryan, Joe

doi:10.1016/j.dam.2014.07.012

Cited by 10 publications

(17 citation statements)

References 4 publications

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“…Next, the papers from each class are separated based on the specific application. The search and selection results are given in Table II for circular and Table III [3], [5], [6], [8][9][10][11][12][13][14][15], [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], [33-42] 37 Total #papers 42 We further categorized the papers from each application in sub-classes depending on whether authors provide formulas on how to calculate the total number of embedded hexagonal cells, their vertices or edges.…”

Section: Paper Search and Selection Resultsmentioning

confidence: 99%

“…A mathematical problem of embedding a hexagonal grid into a bounding hexagon is presented in the literature as a degree diameter problem considering a graph as being embedded in host graph such as honeycomb network. This problem involves finding the largest sub graph in terms of the number of vertices subject to constraints on the degree and the diameter [27]. Diameter D is total number of edges on the circumscribed circle diameter of the hexagon by which the graph is encircled (Fig.…”

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

“…In the former context, a host graph is encircled by a hexagonal ROI with the size given by D. Formulas are developed to find the total number of inner vertices, divided into classes based on the odd and even diameter size [27]:…”

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

“…8 gives an example for both. Authors in [27] do not provide the total number of hexagons embedded inside the ROI. We derive the former with the following formulas:…”

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

“…There is a similar attempt to solve the degree diameter problem while counting the total number of vertices of the subgraph based on the triangular grid which is the basis for hexagonal networks [28]. It is expressed as a function of diameter, defining two classes of architectures, depending on weather the diameter is even or odd:…”

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

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A Review of Embedding Hexagonal Cells in the Circular and Hexagonal Region of Interest

Prvan¹,

Ožegović²,

Mišura³

2019

IJACSA

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Hexagonal cells are applied in various fields of research. They exhibit many advantages, and one of the most important is their possibility to be closely packed and to form a hexagonal grid that fully covers the Region of Interest (ROI) without overlaps or gaps. ROI can be of various geometrical shapes, but this paper deals with the circular or hexagonal ROI approximations. The main purpose of our research is to provide a short review on the literature concerning the hexagonal grid, summarizing the existing state-of-the-art approaches on embedding hexagonal cells in the targeted ROI shapes and offering application-specific advantages. We report on formulas and algebraic expressions given in the existing researches that are used for calculating the number of embedded inner hexagonal cells or their vertices and/or edges. We contribute by integrating all researches in one place, finding a connection between previously unrelated applications concerning the use of embedded hexagonal grid and extracting commonality between previous researches on whether it provides the formulas on calculating the inner hexagon cells. In case only the number of edges or vertices is provided for the targeted application, we derive formulas for calculating the number of inner hexagons. Therefore, our survey results with the overview on solving the problem of embedding hexagonal cells in the desired circular or hexagonal ROI. The contribution of the review is the following: first it provides the existing and the derived formulas for calculating the embedded hexagons and second, it provides a theoretical background that is necessary to encourage further research. Namely, our main motivation, that is the geometrical design of the one of the world's largest CERN particle detectors, Compact Muon Solenoid (CMS) is analyzed as a source for the future research directions.

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Section: Paper Search and Selection Resultsmentioning

confidence: 99%

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

“…8 gives an example for both. Authors in [27] do not provide the total number of hexagons embedded inside the ROI. We derive the former with the following formulas:…”

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

Section: B Roi Approximated By a Hexagon 1) Interconnection Networkmentioning

confidence: 99%

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