The boundary characteristic and Pick's theorem in the Archimedean planar tilings

Ding, Ren; Reay, John R.

doi:10.1016/0097-3165(87)90063-x

Cited by 21 publications

(7 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the orientation of type c, since the path 2 → 7 → 6 → 1 is transitive, P must go through 8 to 1. Thus the subdivision of the cell 17 (13) Thus, e is not a shortcut in this case. If e is a shortcut, then there exists a directed path P of length at least 3 from 1 to 2 (this path does not involve e).…”

Section: Subdivisions Of Triangular Grid Graphs Of General Shapementioning

confidence: 96%

“…The infinite graph T ∞ associated with the two-dimensional triangular grid (also known as the triangular tiling graph, see [13] and [4]) is a graph drawn in the plane with straight-line edges and defined as follows.…”

Section: Triangular Grid Graphsmentioning

confidence: 99%

“…The triangular tiling graph T ∞ (i.e., the two-dimensional triangular grid) is the Archimedean tiling 3 6 is more common introduced in [13] and [4]. By a triangular grid graph G in this paper we mean a graph obtained from T ∞ as follows.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Word-Representability of Face Subdivisions of Triangular Grid Graphs

Chen

Kitaev

Sun

2016

Graphs and Combinatorics

View full text Add to dashboard Cite

This version is available at https://strathprints.strath.ac.uk/55889/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Inspired by a recent elegant result of Akrobotu et al., who classified wordrepresentable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpiński gasket graph is wordrepresentable.

show abstract

Section: Subdivisions Of Triangular Grid Graphs Of General Shapementioning

confidence: 96%

Section: Triangular Grid Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Word-Representability of Face Subdivisions of Triangular Grid Graphs

Chen

Kitaev

Sun

2016

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

“…For a discussion of alternative approaches see the article by Niven and Zuckermann [15]. There are numerous generalizations to other than the orthogonal grid, e.g., by Ren and Reay [23]; see [22] for a generalization to higher dimensions.…”

Section: Pick's Theoremmentioning

confidence: 99%

On Simple Polygonalizations with Optimal Area

Fekete

2000

Discrete Comput Geom

View full text Add to dashboard Cite

We discuss the problem of finding a simple polygonalization for a given set of vertices P that has optimal area. We show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P and prove that it is NP-complete to find a minimum weight polygon or a maximum weight polygon for a given vertex set, resulting in a proof of NP-completeness for the corresponding area optimization problems. This answers a generalization of a question stated by Suri in 1989. Finally, we turn to higher dimensions, where we prove that, for 1 ≤ k ≤ d, 2 ≤ d, it is NP-hard to determine the smallest possible total volume of the k-dimensional faces of a d-dimensional simple nondegenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980.

show abstract

“…Since 1960, many papers have been published concerning Pick's formula. They contain several proofs of the formula [1,3,4,11,12,16,20,27,28,29] or proof of the equivalence of this result with other ones [5,7,8,15] or generalizations to more general polygons [2,6,9,10,24,25,26,30] to more general lattices [19,23] and also to higher-dimensional polyhedra [13,21,22,25].…”

Section: Introductionmentioning

confidence: 99%

Euler's characteristics and Pick's theorem

Dubeau¹,

Labbé²

2007

ijcms

View full text Add to dashboard Cite

Pick's Theorem is used to compute the area of lattice polygons. In this paper we present a very general definition of a polygon to obtain the definition of faces and holes of a polygon. The decomposition of a polygon into triangles is briefly discussed. Then lattices are introduced with their corresponding elementary triangles and triangulations. We recall and prove Pick's Theorem for simple polygons. Then a generalization of Pick's Theorem is proved for very general lattice polygons, generalized lattice polygons and union of generalized lattice polygons using original and modified Euler's characteristics.

show abstract

The boundary characteristic and Pick's theorem in the Archimedean planar tilings

Cited by 21 publications

References 8 publications

Word-Representability of Face Subdivisions of Triangular Grid Graphs

Word-Representability of Face Subdivisions of Triangular Grid Graphs

On Simple Polygonalizations with Optimal Area

Euler's characteristics and Pick's theorem

Contact Info

Product

Resources

About