An Introduction to Intersection Graphs

2020 ◽  
pp. 24-65 ◽  
Author(s):  
Madhumangal Pal
Keyword(s):  
Interval Graphs ◽  
Chordal Graphs ◽  
Intersection Graph ◽  
Important Class ◽  
Intersection Graphs ◽  
Permutation Graphs ◽  
Circular Arc ◽  
String Graphs ◽  
Circular Arc Graphs ◽  
Trapezoid Graphs

In this chapter, a very important class of graphs called intersection graph is introduced. Based on the geometrical representation, many different types of intersection graphs can be defined with interesting properties. Some of them—interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, chordal graphs, line graphs, disk graphs, string graphs—are presented here. A brief introduction of each of these intersection graphs along with some basic properties and algorithmic status are investigated.

Efficient algorithms for interval graphs and circular-arc graphs

Networks ◽  
1982 ◽  
Vol 12 (4) ◽  
pp. 459-467 ◽  
Author(s):  
U. I. Gupta ◽  
D. T. Lee ◽  
J. Y.-T. Leung
Keyword(s):  
Efficient Algorithms ◽  
Interval Graphs ◽  
Circular Arc ◽  
Circular Arc Graphs

scholarly journals Proper circular arc graphs as intersection graphs of pathson a grid

2019 ◽  
Vol 262 ◽  
pp. 195-202 ◽  
Author(s):  
Esther Galby ◽  
María Pía Mazzoleni ◽  
Bernard Ries
Keyword(s):  
Intersection Graphs ◽  
Circular Arc ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

L(h,k)-Labeling of Intersection Graphs

2020 ◽  
pp. 135-170
Author(s):  
Sk. Amanathulla ◽  
Madhumangal Pal
Keyword(s):  
Coding Theory ◽  
Chordal Graph ◽  
Interval Graph ◽  
Graph Labeling ◽  
Frequency Assignment ◽  
Permutation Graph ◽  
Intersection Graphs ◽  
Circular Arc ◽  
Trapezoid Graph ◽  
Circular Arc Graphs

One important problem in graph theory is graph coloring or graph labeling. Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in (mobile) communication system, coding theory, ray crystallography, radar, circuit design, etc. For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers such that if and if, where it represents the distance between the nodes. Intersection graph is a very important subclass of graph. Unit disc graph, chordal graph, interval graph, circular-arc graph, permutation graph, trapezoid graph, etc. are the important subclasses of intersection graphs. In this chapter, the authors discuss labeling for intersection graphs, specially for interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, etc., and have presented a lot of results for this problem.

scholarly journals Novel Procedures for Graph Edge-colouring

2019 ◽  
Author(s):  
Leandro M. Zatesko ◽  
Renato Carmo ◽  
André L. P. Guedes
Keyword(s):  
Uniform Distribution ◽  
Polynomial Time ◽  
Chordal Graphs ◽  
Circular Arc ◽  
Simple Graphs ◽  
Degree Sum ◽  
Local Degree ◽  
Polynomial Time Algorithms ◽  
Circular Arc Graphs ◽  
Edge Colouring

We present a novel recolouring procedure for graph edge-colouring. We show that all graphs whose vertices have local degree sum not too large can be optimally edge-coloured in polynomial time. We also show that the set ofthe graphs satisfying this condition includes almost every graph (under the uniform distribution). We present further results on edge-colouring join graphs, chordal graphs, circular-arc graphs, and complementary prisms, whose proofs yield polynomial-time algorithms. Our results contribute towards settling the Over- full Conjecture, the main open conjecture on edge-colouring simple graphs. Fi- nally, we also present some results on total colouring.

scholarly journals On Total and Edge-colouring of Proper Circular-arc Graphs

2018 ◽  
Author(s):  
João Pedro W. Bernardi ◽  
Sheila M. De Almeida ◽  
Leandro M. Zatesko
Keyword(s):  
Linear Time ◽  
Sufficient Conditions ◽  
Interval Graphs ◽  
Circular Arc ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Np Complete ◽  
Polynomially Solvable ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Deciding if a graph is Δ-edge-colourable (resp. (Δ + 1)-total colourable), although it is an NP-complete problem for graphs in general, is polynomially solvable for interval graphs of odd (resp. even) maximum degree Δ. An interesting superclass of the proper interval graphs are the proper circular-arc graphs, for which we suspect that Δ-edge-colourability is linear-time decidable. This work presents sufficient conditions for Δ-edge-colourability, (Δ + 1)-total colourability, and (Δ+2)-total colourability of proper circular-arc graphs. Our proofs are constructive and yield polynomial-time algorithms.

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