scholarly journals Even-hole-free graphs: A survey

2010 ◽  
Vol 4 (2) ◽  
pp. 219-240 ◽  
Author(s):  
Kristina Vuskovic
Keyword(s):  
Polynomial Time ◽  
Perfect Graphs ◽  
Independent Interest ◽  
Perfect Graph ◽  
Graph Classes ◽  
Structural Characterizations

The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of even-hole-free graphs were generalized to other complex hereditary graph classes, and in the case of perfect graphs led to the famous resolution of the Strong Perfect Graph Conjecture and their polynomial time recognition. The class of even-holefree graphs is also of independent interest due to its relationship to ?-perfect graphs. In this survey we describe all the different structural characterizations of even-hole-free graphs, focusing on their algorithmic consequences.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

2000 ◽  
Vol 11 (03) ◽  
pp. 423-443 ◽  
Author(s):  
MARTIN CHARLES GOLUMBIC ◽  
UDI ROTICS
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Unit Interval ◽  
Perfect Graphs ◽  
The Other ◽  
Perfect Graph ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Unit Interval Graph

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

scholarly journals Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

scholarly journals Parameterized Complexity of Vertex Deletion into Perfect Graph Classes

2011 ◽  
pp. 240-251 ◽  
Author(s):  
Pinar Heggernes ◽  
Pim van’t Hof ◽  
Bart M. P. Jansen ◽  
Stefan Kratsch ◽  
Yngve Villanger
Keyword(s):  
Parameterized Complexity ◽  
Perfect Graph ◽  
Graph Classes ◽  
Vertex Deletion

scholarly journals Parameterized complexity of vertex deletion into perfect graph classes

2013 ◽  
Vol 511 ◽  
pp. 172-180 ◽  
Author(s):  
Pinar Heggernes ◽  
Pim van ’t Hof ◽  
Bart M.P. Jansen ◽  
Stefan Kratsch ◽  
Yngve Villanger
Keyword(s):  
Parameterized Complexity ◽  
Perfect Graph ◽  
Graph Classes ◽  
Vertex Deletion

Computing clique and chromatic number of circular-perfect graphs in polynomial time

2012 ◽  
Vol 141 (1-2) ◽  
pp. 121-133 ◽  
Author(s):  
Arnaud Pêcher ◽  
Annegret K. Wagler
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Perfect Graphs

scholarly journals On the Helly Property of Some Intersection Graphs

2021 ◽  
Author(s):  
Tanilson D. Santos ◽  
Jayme Szwarcfiter ◽  
Uéverton S. Souza ◽  
Claudson F. Bornstein
Keyword(s):  
Intersection Graph ◽  
Negative Integer ◽  
Intersection Graphs ◽  
Graph Classes ◽  
Intersection Model ◽  
Helly Property ◽  
Np Complete ◽  
Structural Characterizations

An EPG graph G is an edge-intersection graph of paths on a grid. In this thesis, we analyze structural characterizations and complexity aspects regarding EPG graphs. Our main focus is on the class of B1-EPG graphs whose intersection model satisfies well-known the Helly property, called Helly-B1-EPG. We show that the problem of recognizing Helly-B1-EPG graphs is NP-complete. Besides, other intersection graph classes such as VPG, EPT, and VPT were also studied. We completely solve the problem of determining the Helly and strong Helly numbers of Bk-EPG graphs and Bk-VPG graphs for each non-negative integer k. Finally, we show that every Chordal B1-EPG graph is at the intersection of VPT and EPT.

Applications of Super Strongly Perfect Graph for Manufacturing System towards a Leaner Structure

2015 ◽  
Vol 766-767 ◽  
pp. 943-948
Author(s):  
R. Mary Jeya Jothi
Keyword(s):  
Graph Theory ◽  
Structural Characterization ◽  
Manufacturing System ◽  
Dominating Set ◽  
Perfect Graphs ◽  
Perfect Graph ◽  
System Models ◽  
Wheel Graphs ◽  
Minimal Dominating Set ◽  
Original Configuration

Some restructuring decisions are conceptualized which reflect the aim of the organization to gradually evolve the manufacturing system towards a leaner structure. This is done by way of defining simplified process so that lesser hindrance in terms of cycles of interactions is found. The reframing decisions are given by five restructured configurations of the manufacturing system. Models using graph theory are developed for original configuration and each of the new reframed configurations and the resulting structural characterization information is used to compare the structure of restructured configurations with the original configuration. A graph G is Super Strongly Perfect (SSP) if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H. A study on some classes of super strongly perfect graphs like wheel and double wheel graphs (in which each graph represents structure of some manufacturing system) are given.

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