Shortest paths in intersection graphs of unit disks

A conflict-free[Formula: see text]-coloring of a graph [Formula: see text] assigns one of [Formula: see text] different colors to some of the vertices such that, for every vertex [Formula: see text], there is a color that is assigned to exactly one vertex among [Formula: see text] and [Formula: see text]’s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of [Formula: see text] geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in [Formula: see text] and in [Formula: see text] for disks or squares of different sizes; it is known for general graphs that the worst case is in [Formula: see text]. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.

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eMap: A Web Application for Identifying and Visualizing Electron or Hole Hopping Pathways in Proteins

10.26434/chemrxiv.7889993 ◽

2019 ◽

Author(s):

Ruslan N. Tazhigulov ◽

James R. Gayvert ◽

Melissa Wei ◽

Ksenia B. Bravaya

Keyword(s):

Web Application ◽

Shortest Paths ◽

Coarse Grained ◽

Decay Parameter ◽

Electron Hole ◽

Web Based ◽

Aromatic Amino Acid Residue ◽

Pathways Model ◽

Visualization Tools ◽

Effective Decay

<p>eMap is a web-based platform for identifying and visualizing electron or hole transfer pathways in proteins based on their crystal structures. The underlying model can be viewed as a coarse-grained version of the Pathways model, where each tunneling step between hopping sites represented by electron transfer active (ETA) moieties is described with one eﬀective decay parameter that describes protein-mediated tunneling. ETA moieties include aromatic amino acid residue side chains and aromatic fragments of cofactors that are automatically detected, and, in addition, electron/hole residing sites that can be speciﬁed by the users. The software searches for the shortest paths connecting the user-speciﬁed electron/hole source to either all surface-exposed ETA residues or to the user-speciﬁed target. The identiﬁed pathways are ranked based on their length. The pathways are visualized in 2D as a graph, in which each node represents an ETA site, and in 3D using available protein visualization tools. Here, we present the capability and user interface of eMap 1.0, which is available at https://emap.bu.edu.</p>

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Computer algorithms

10.1093/oso/9780198805090.003.0008 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Data Structures ◽

Analysis Of Algorithms ◽

Shortest Paths ◽

Minimum Cut ◽

Network Data ◽

Computer Algorithms ◽

Breadth First Search ◽

Augmenting Path ◽

Basic Concepts ◽

Maximum Flows

This chapter introduces some of the fundamental concepts of numerical network calculations. The chapter starts with a discussion of basic concepts of computational complexity and data structures for storing network data, then progresses to the description and analysis of algorithms for a range of network calculations: breadth-first search and its use for calculating shortest paths, shortest distances, components, closeness, and betweenness; Dijkstra's algorithm for shortest paths and distances on weighted networks; and the augmenting path algorithm for calculating maximum flows, minimum cut sets, and independent paths in networks.

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Lower bounds for computing geometric spanners and approximate shortest paths

Discrete Applied Mathematics ◽

10.1016/s0166-218x(00)00280-8 ◽

2001 ◽

Vol 110 (2-3) ◽

pp. 151-167 ◽

Cited By ~ 12

Author(s):

Danny Z. Chen ◽

Gautam Das ◽

Michiel Smid

Keyword(s):

Lower Bounds ◽

Shortest Paths ◽

Geometric Spanners

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Computational Algorithms for Censored-Data Problems Using Intersection Graphs

Journal of Computational and Graphical Statistics ◽

10.1198/106186001317114901 ◽

2001 ◽

Vol 10 (3) ◽

pp. 403-421 ◽

Cited By ~ 31

Author(s):

R Gentleman ◽

A. C Vandal

Keyword(s):

Censored Data ◽

Computational Algorithms ◽

Intersection Graphs

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Turán-type results for intersection graphs of boxes

Combinatorics Probability Computing ◽

10.1017/s0963548321000171 ◽

2021 ◽

pp. 1-6

Author(s):

István Tomon ◽

Dmitriy Zakharov

Keyword(s):

Short Note ◽

Intersection Graph ◽

Intersection Graphs ◽

Poset Dimension ◽

Axis Parallel ◽

Turán Theorem ◽

Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

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