scholarly journals Conflict-Free Coloring of Intersection Graphs

2018 ◽  
Vol 28 (03) ◽  
pp. 289-307 ◽  
Author(s):  
Sándor P. Fekete ◽  
Phillip Keldenich
Keyword(s):  
Graph Theory ◽  
Unit Disk ◽  
Wireless Networking ◽  
Worst Case ◽  
Intersection Graphs ◽  
Geometric Objects ◽  
Np Complete ◽  
Geometric Intersection Graphs ◽  
Unit Disks ◽  
General Graphs

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.

ROMAN DOMINATION AND ITS VARIANTS IN UNIT DISK GRAPHS

2010 ◽  
Vol 02 (01) ◽  
pp. 99-105 ◽  
Author(s):  
WEIPING SHANG ◽  
XIUMEI WANG ◽  
XIAODONG HU
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Wireless Ad Hoc Networks ◽  
Ad Hoc ◽  
Dominating Set ◽  
Roman Domination ◽  
Intersection Graphs ◽  
Unit Disk Graphs ◽  
Np Complete ◽  
Hoc Networks

Unit disk graphs are the intersection graphs of equal sized disks in the plane, they are widely used as a mathematical model for wireless ad-hoc networks and some problems in computational geometry. In this paper we first show that Roman dominating set and connected Roman dominating set problems in unit disk graphs are NP-complete, and then present two approximation algorithms for these problems.

Online Coloring and $L(2,1)$-Labeling of Unit Disk Intersection Graphs

2018 ◽  
Vol 32 (2) ◽  
pp. 1335-1350 ◽  
Author(s):  
Konstanty Junosza-Szaniawski ◽  
Paweł Rzaͅżewski ◽  
Joanna Sokół ◽  
Krzysztof Weͅsek
Keyword(s):  
Unit Disk ◽  
Intersection Graphs ◽  
Online Coloring

Framed 4-valent graph minor theory I: Introduction. A planarity criterion and linkless embeddability

2014 ◽  
Vol 23 (07) ◽  
pp. 1460002
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Graph Theory ◽  
The Other ◽  
Graph Minor ◽  
Other Hand ◽  
Simple Class ◽  
Points Of View ◽  
General Graphs ◽  
Planarity Criterion

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this paper is to collect some well-known results on planarity and to reformulate them in the language of minors. The goal of the whole sequence is to prove analogs of the Robertson–Seymour–Thomas theorems for framed 4-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed 4-graphs are easier to consider than general graphs; on the other hand, framed 4-graphs are closely related to many problems in graph theory.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

scholarly journals Covering random points in a unit disk

2008 ◽  
Vol 40 (01) ◽  
pp. 22-30
Author(s):  
Jennie C. Hansen ◽  
Eric Schmutz ◽  
Li Sheng
Keyword(s):  
Uniform Distribution ◽  
Unit Disk ◽  
Asymptotic Probability ◽  
Unit Disks

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .

A O(m) Self-Stabilizing Algorithm for Maximal Triangle Partition of General Graphs

2017 ◽  
Vol 27 (02) ◽  
pp. 1750004
Author(s):  
Brahim Neggazi ◽  
Volker Turau ◽  
Mohammed Haddad ◽  
Hamamache Kheddouci
Keyword(s):  
Graph Matching ◽  
Partition Problem ◽  
Matching Problem ◽  
Np Complete ◽  
General Graphs

The triangle partition problem is a generalization of the well-known graph matching problem consisting of finding the maximum number of independent edges in a given graph, i.e., edges with no common node. Triangle partition instead aims to find the maximum number of disjoint triangles. The triangle partition problem is known to be NP-complete. Thus, in this paper, the focus is on the local maximization variant, called maximal triangle partition (MTP). Thus, paper presents a new self-stabilizing algorithm for MTP that converges in O(m) moves under the unfair distributed daemon.

CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS

2013 ◽  
Vol 05 (04) ◽  
pp. 1350024 ◽  
Author(s):  
B. S. PANDA ◽  
S. PAUL
Keyword(s):  
Dominating Set ◽  
Chordal Graphs ◽  
Hardness Of Approximation ◽  
Induced Subgraph ◽  
Path Graph ◽  
Block Graphs ◽  
Domination Problem ◽  
Np Complete ◽  
Domination In Graphs ◽  
General Graphs

A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs.

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