scholarly journals Boxicity, Poset Dimension, and Excluded Minors

10.37236/7787 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Louis Esperet ◽  
Veit Wiechert
Keyword(s):  
Short Note ◽  
Interval Graphs ◽  
Circular Arc ◽  
Poset Dimension ◽  
Circular Arc Graphs ◽  
Excluded Minors

In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.

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

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.

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.

scholarly journals Turán-type results for intersection graphs of boxes

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.

Surjective L (2, 1)-labeling of cycles and circular-arc graphs

2018 ◽  
Vol 35 (1) ◽  
pp. 739-748 ◽  
Author(s):  
Sk. Amanathulla ◽  
Madhumangal Pal
Keyword(s):  
Circular Arc ◽  
Circular Arc Graphs
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