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 Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs

2015 ◽  
Vol 115 (2) ◽  
pp. 256-262 ◽  
Author(s):  
Andreas Brandstädt ◽  
Pavel Fičur ◽  
Arne Leitert ◽  
Martin Milanič
Keyword(s):  
Polynomial Time ◽  
Chordal Graphs ◽  
Polynomial Time Algorithms ◽  
Domination Problems

Polynomial time recognition of unit circular-arc graphs

2006 ◽  
Vol 58 (1) ◽  
pp. 67-78 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Ross M. McConnell ◽  
Jeremy Spinrad ◽  
Alan Tucker
Keyword(s):  
Polynomial Time ◽  
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 Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems

2018 ◽  
Author(s):  
Bhadrachalam Chitturi ◽  
Srijith Balachander ◽  
Sandeep Satheesh ◽  
Krithic Puthiyoppil
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Vertex Cover ◽  
Independent Set ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Graph Class ◽  
Hard Problems ◽  
Connected Vertex Cover ◽  
Circular Arc Graphs

The independent set, IS, on a graph G = ( V , E ) is V * ⊆ V such that no two vertices in V * have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. V * ⊆ V is a vertex cover, i.e. VC of G = ( V , E ) if every e ∈ E is incident upon at least one vertex in V * . V * ⊆ V is dominating set, DS, of G = ( V , E ) if ∀ v ∈ V either v ∈ V * or ∃ u ∈ V * and ( u , v ) ∈ E . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if k = Θ ( log ∣ V ∣ ) then MIS, MVC and MDS can be computed in polynomial time and if k = O ( ( log ∣ V ∣ ) α ) , where α < 1 , then MCV and MCD can be computed in polynomial time. If k = Θ ( ( log ∣ V ∣ ) 1 + ϵ ) , for ϵ > 0 , then MIS, MVC and MDS require quasi-polynomial time. If k = Θ ( log ∣ V ∣ ) then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.

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.

Polynomial time algorithms on circular-arc overlap graphs

Networks ◽  
1991 ◽  
Vol 21 (2) ◽  
pp. 195-203 ◽  
Author(s):  
Toshinobu Kashiwabara ◽  
Sumio Masuda ◽  
Kazuo Nakajima ◽  
Toshio Fujisawa
Keyword(s):  
Polynomial Time ◽  
Circular Arc ◽  
Polynomial Time Algorithms ◽  
Overlap Graphs
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