scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

scholarly journals Complexity of domination in triangulated plane graphs

2019 ◽  
Vol 11 (2) ◽  
pp. 174-183
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Domination Number ◽  
Plane Graph ◽  
Plane Graphs ◽  
Power Domination ◽  
Np Complete

Abstract We prove that for a triangulated plane graph it is NP-complete to determine its domination number and its power domination number.

GENERA OF THE LINKS DERIVED FROM 2-CONNECTED PLANE GRAPHS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250129 ◽  
Author(s):  
SHUYA LIU ◽  
HEPING ZHANG
Keyword(s):  
Explicit Formula ◽  
Large Family ◽  
Plane Graph ◽  
Plane Graphs ◽  
Degree Sum ◽  
Oriented Link

In this paper, we associate a plane graph G with an oriented link by replacing each vertex of G with a special oriented n-tangle diagram. It is shown that such an oriented link has the minimum genus over all orientations of its unoriented version if its associated plane graph G is 2-connected. As a result, the genera of a large family of unoriented links are determined by an explicit formula in terms of their component numbers and the degree sum of their associated plane graphs.

scholarly journals A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Marcus Schaefer
Keyword(s):  
Plane Graph ◽  
Plane Graphs ◽  
Classic Result

We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge. This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.

scholarly journals NP-Completeness of st-orientations for plane graphs

2010 ◽  
Vol 411 (7-9) ◽  
pp. 995-1003 ◽  
Author(s):  
Sadish Sadasivam ◽  
Huaming Zhang
Keyword(s):  
Plane Graphs ◽  
Np Completeness

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

COMPLEXITY AND APPROXIMABILITY OF DOUBLE DIGEST

2005 ◽  
Vol 03 (02) ◽  
pp. 207-223
Author(s):  
MARK CIELIEBAK ◽  
STEPHAN EIDENBENZ ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Real Life ◽  
Approximation Ratio ◽  
Additional Restriction ◽  
Np Hard ◽  
Model Errors ◽  
Np Completeness ◽  
The Third ◽  
Np Complete ◽  
Feasible Solutions

We revisit the DOUBLE DIGEST problem, which occurs in sequencing of large DNA strings and consists of reconstructing the relative positions of cut sites from two different enzymes. We first show that DOUBLE DIGEST is strongly NP-complete, improving upon previous results that only showed weak NP-completeness. Even the (experimentally more meaningful) variation in which we disallow coincident cut sites turns out to be strongly NP-complete. In the second part, we model errors in data as they occur in real-life experiments: we propose several optimization variations of DOUBLE DIGEST that model partial cleavage errors. We then show that most of these variations are hard to approximate. In the third part, we investigate variations with the additional restriction that coincident cut sites are disallowed, and we show that it is NP-hard to even find feasible solutions in this case, thus making it impossible to guarantee any approximation ratio at all.

DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO LATTICES

2009 ◽  
Vol 18 (12) ◽  
pp. 1711-1726 ◽  
Author(s):  
XIAN'AN JIN ◽  
FENGMING DONG ◽  
ENG GUAN TAY
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Plane Graph ◽  
Link Diagram ◽  
Plane Graphs ◽  
Link Diagrams ◽  
One To One

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

On the Hardness of Devising Interval Routing Schemes

1997 ◽  
Vol 07 (01) ◽  
pp. 39-47 ◽  
Author(s):  
Michele Flammini
Keyword(s):  
Time Complexity ◽  
Compact Routing ◽  
Routing Schemes ◽  
Np Completeness ◽  
Routing Scheme ◽  
Np Complete ◽  
Interval Routing ◽  
General Networks

The k-Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation of the INMOS transputer router chips. In this paper we investigate the time complexity of devising a minimal space k-IRS and we prove that the problem of deciding whether there exists a 2-IRS for any network G is NP-complete. This is the first hardness result for k-IRS where k is constant and the graph underlying the network is unweighted. Moreover, the NP-completeness holds also for linear and strict 2-IRS.

scholarly journals On subset sum problem in branch groups

2020 ◽  
Vol Volume 12, issue 1 ◽  
Author(s):  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Final Version ◽  
Subset Sum Problem ◽  
Grigorchuk Group ◽  
Np Completeness ◽  
Subset Sum ◽  
Np Complete ◽  
Np Hardness

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting. Comment: v3: final version for journal of Groups, Complexity, Cryptology. arXiv admin note: text overlap with arXiv:1703.07406

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