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 Complexity of Hamiltonian Cycle Reconfiguration

Algorithms ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 140 ◽  
Author(s):  
Asahi Takaoka
Keyword(s):  
Hamiltonian Cycle ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Interval Graph ◽  
Unit Interval ◽  
Chordal Graphs ◽  
Hamiltonian Cycles ◽  
Permutation Graph ◽  
Proper Subclass ◽  
Chordal Bipartite Graphs

The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

scholarly journals Classes of Graphs with $e$-Positive Chromatic Symmetric Function

10.37236/8211 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Angèle M. Foley ◽  
Chính T. Hoàng ◽  
Owen D. Merkel
Keyword(s):  
Symmetric Functions ◽  
Interval Graph ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Related Case ◽  
Graph Classes ◽  
Unit Interval Graph ◽  
Related Graph ◽  
Chromatic Symmetric Function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove  that unit interval graphs whose complement is also a unit interval graph are $e$-positive.   We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.  

BIPARTITE PERMUTATION GRAPHS ARE RECONSTRUCTIBLE

2012 ◽  
Vol 04 (03) ◽  
pp. 1250039
Author(s):  
MASASHI KIYOMI ◽  
TOSHIKI SAITOH ◽  
RYUHEI UEHARA
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Unit Interval ◽  
Regular Graphs ◽  
Pendant Vertex ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Graph Reconstruction ◽  
Bipartite Permutation Graphs ◽  
Disconnected Graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. The conjecture has been verified for all graphs with at most 11 vertices. Further, the conjecture has been verified for regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs with no pendant vertex, outer-planar graphs, and unicyclic graphs. We extend the list of graph classes for which the conjecture holds. We give a proof that bipartite permutation graphs are reconstructible.

scholarly journals A Structural Characterization for Certifying Robinsonian Matrices

10.37236/6701 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Monique Laurent ◽  
Matteo Seminaroti ◽  
Shin-ichi Tanigawa
Keyword(s):  
Structural Characterization ◽  
Adjacency Matrix ◽  
Efficient Algorithm ◽  
Symmetric Matrix ◽  
Interval Graph ◽  
Unit Interval ◽  
Interval Graphs ◽  
Weighted Graphs ◽  
Unit Interval Graph

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of  asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian.

scholarly journals A four-sweep LBFS recognition algorithm for interval graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Recognition Algorithm ◽  
Structure Theory ◽  
Time Interval ◽  
Breadth First Search ◽  
Input Graph ◽  
Graph Recognition ◽  
Unit Interval Graph ◽  
International Audience

Graph Theory International audience In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

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.

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

Further Study on Reverse 1-Center Problem on Trees

2020 ◽  
Vol 37 (06) ◽  
pp. 2050034
Author(s):  
Ali Reza Sepasian ◽  
Javad Tayyebi
Keyword(s):  
Dynamic Programming ◽  
Time Complexity ◽  
Numerical Experiments ◽  
Linear Time ◽  
Minimum Cost ◽  
Fixed Number ◽  
The Other ◽  
Center Problem ◽  
Objective Value ◽  
Linear Cost

This paper studies two types of reverse 1-center problems under uniform linear cost function where edge lengths are allowed to reduce. In the first type, the aim is that the objective value is bounded by a prescribed fixed value [Formula: see text] at minimum cost. The aim of the other is to improve the objective value as much as possible within a given budget. An algorithm based on dynamic programming is proposed to solve the first problem in linear time. Then, this algorithm is applied as a subroutine to design an algorithm to solve the second type of the problem in [Formula: see text] time in which [Formula: see text] is a fixed number dependent on the problem parameters. Under the similarity assumption, this algorithm has a better complexity than the Nguyen algorithm (2013) with quadratic-time complexity. Some numerical experiments are conducted to validate this fact in practice.

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