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List Partitions of Chordal Graphs
[chapter]

2004
*
Lecture Notes in Computer Science
*

We show that many Mpartition problems that are NP-complete in general become solvable in polynomial time for

doi:10.1007/978-3-540-24698-5_14
fatcat:fb7vjcua3rcetplb4jqzpgh4xa
*chordal**graphs*, even in the presence*of**lists*. ... In this paper we expand our focus and consider general M -*partitions*, also known as trigraph homomorphisms, for the class*of**chordal**graphs*. ... In this paper, we consider the restrictions*of*both the M -*partition*and the*list*M -*partition*problems to instances G that are*chordal**graphs*. ...##
###
List matrix partitions of chordal graphs

2005
*
Theoretical Computer Science
*

Such (

doi:10.1016/j.tcs.2005.09.030
fatcat:qnola5hxpjdb3go5wd2ia5zu44
*list*)*partitions*generalize (*list*) colourings and (*list*) homomorphisms, and arise frequently in the theory*of**graph*perfection. ... Both*of*these problems are in fact*partition*problems, restricted to*chordal**graphs*. ... To see that such a*chordal**graph*G 0 with*lists*exits, suppose that the*list*M-*partition*problem is NP-complete but every*chordal**graph*G with*lists*admits a*list*M-*partition*. ...##
###
Polarity of chordal graphs

2008
*
Discrete Applied Mathematics
*

They are defined by the existence

doi:10.1016/j.dam.2008.01.026
fatcat:qjcpdwgb5bfu3hj6ddraft2fpa
*of*a certain*partition**of*vertices, which is NP-complete to decide for general*graphs*. ... In this paper we address the question*of*polarity*of**chordal**graphs*, arguing that this is in essence a question*of*colourability, and hence*chordal**graphs*are a natural restriction. ... A monopolar*partition**of*a*chordal**graph*is a split*partition*if and only if there is exactly one blue clique. As noted above, we can obtain all maximal cliques*of*a*chordal**graph*in linear time. ...##
###
Minimal obstructions for a matrix partition problem in chordal graphs
[article]

2020
*
arXiv
*
pre-print

If M is an m × m matrix over { 0, 1, ∗}, an M-

arXiv:2004.01229v1
fatcat:mol265vseffj7p2nalkdqwweom
*partition**of*a*graph*G is a*partition*(V_1, ... ... It is known that for every 3 × 3 matrix M over { 0, 1, ∗}, there are finitely many*chordal*minimal obstructions for the problem*of*determining whether a*graph*admits an M-*partition*, except for two matrices ... a*partition*into k independent sets and cliques, [5] considers the*list*version*of*the problem on*chordal**graphs*, polarity*of**chordal**graphs*is studied in [3] , in [10] the M -*partition*problem is ...##
###
The P versus NP–complete dichotomy of some challenging problems in graph theory

2012
*
Discrete Applied Mathematics
*

Some open problems are discussed: the stubborn problem for

doi:10.1016/j.dam.2010.12.014
fatcat:vou3y5m4izgzpcgze6ocw6llrm
*list**partition*, the chromatic index*of**chordal**graphs*, and the recognition*of*split clique*graphs*. ... We shall discuss our contribution through the classification*of*some long-standing problems in important areas*of**graph*theory: perfect*graphs*, intersection*graphs*, and structural characterization*of**graph*... Ferreira and to Fabio Protti, the Program Chairs*of*LAGOS 2009, for the opportunity*of*giving this invited talk. Special thanks are due to my co-authors. ...##
###
Obstructions to partitions of chordal graphs

2013
*
Discrete Mathematics
*

We discuss these problems for the class

doi:10.1016/j.disc.2012.05.023
fatcat:hsybgobk7vh57eojttaq472fwm
*of**chordal**graphs*. ... Matrix*partition*problems generalize*graph*colouring and homomorphism problems, and occur frequently in the study*of*perfect*graphs*. ... In the*list*variant, the vertices*of*the input*graph*G have*lists*(*of*allowed parts), and an Mpartition must place each vertex*of*G in a part that is allowed for it [2, 12] . ...##
###
Star Partitions of Perfect Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

*graphs*, and also NP-hard cases, for example, on grid

*graphs*and

*chordal*

*graphs*. ... The

*partition*

*of*

*graphs*into nice subgraphs is a central algorithmic problem with strong ties to matching theory. ... We show that P 3 -

*Partition*restricted to

*chordal*

*graphs*is NP-hard by reduction from 3-Dimensional Matching. Theorem 7. P 3 -

*Partition*restricted to

*chordal*

*graphs*is NP-hard. ...

##
###
Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing

2000
*
Theoretical Computer Science
*

By making use

doi:10.1016/s0304-3975(97)00241-7
fatcat:brlhuthj7zdmfib5ksrfcp6t2m
*of*lexicographic breadth rst search (Lex-BFS) and*partition*re nement with pivots, we obtain very simple algorithms for some well-known problems in*graph*theory. ... We give an O(n + m log n) algorithm for transitive orientation*of*a comparability*graph*, and simple linear algorithms to recognize interval*graphs*, convex*graphs*, Y -semichordal*graphs*and matrices that ... matrix*of*a*chordal**graph*, in O(r + c + m) time. ...##
###
Partitioning chordal graphs into independent sets and cliques

2004
*
Discrete Applied Mathematics
*

The algorithm and the characterization extend, in a natural way, to the '

doi:10.1016/s0166-218x(03)00371-8
fatcat:nscz7zaevbakzkl3jp2wdwshby
*list*' (or 'pre-colouring extension') version*of*the split*partition*problem-given a*graph*with some vertices pre-assigned to the ... We consider the following generalization*of*split*graphs*: A*graph*is said to be a (k; ')-*graph*if its vertex set can be*partitioned*into k independent sets and ' cliques. ... Feder, will appear in our paper '*List*matrix*partitions**of**chordal**graphs*'. Theorem 7 . 7 A*chordal**graph*is a (k; ')-*graph*if and only if it does not contain (' + 1)K k+1 as an induced subgraph. ...##
###
From a simple elimination ordering to a strong elimination ordering in linear time

2003
*
Information Processing Letters
*

We present a linear time algorithm for transforming a simple elimination ordering

doi:10.1016/s0020-0190(03)00228-x
fatcat:k6wlwyeqfncrfnwynxad3bwwvq
*of*a strongly*chordal**graph*into a strong elimination ordering. ... Function MakeSets (G: strongly*chordal**graph*, [v 1 , . . . , v n ]: simple elimination ordering) returns ordered*list**of*sets that*partition*V ; (1) L := empty*list**of*sets; Lemma 1 . 1 If L = S 1 , ... . , v n*of*a strongly*chordal**graph*G. The following function*partitions*the vertices into a*list*L*of*disjoint sets. (2) for t := 1 to n do begin a, b, g, c, d, e, f . ...##
###
Matrix Partitions of Split Graphs
[article]

2013
*
arXiv
*
pre-print

Matrix

arXiv:1306.1967v2
fatcat:wbzw246jkffkdp4sj6axkjpiee
*partition*problems generalize a number*of*natural*graph**partition*problems, and have been studied for several standard*graph*classes. ... (In particular, there are matrix*partition*problems which have infinitely many minimal*chordal*obstructions.) ... When dealing with the M-*partition*problem with*lists*, it is shown in [1] that there is a matrix M for which the*list*M-*partition*problem is NP -complete, even when restricted to*chordal**graphs*. ...##
###
Hardness Results and Efficient Algorithms for Graph Powers
[chapter]

2010
*
Lecture Notes in Computer Science
*

Discrete Math. 18 (2004) 83-102] proved that recognizing squares

doi:10.1007/978-3-642-11409-0_21
fatcat:eqwta4d65rg43pr6o4grlfyllq
*of**chordal**graphs*and squares*of*split*graphs*are NP-complete. ... Lau and Corneil [Recognizing powers*of*proper interval, split and*chordal**graphs*, SIAM J. ... A split*graph*is one whose vertex set can be*partitioned*into a clique and a stable set. Clearly, split*graphs*are*chordal*. ...##
###
On listing, sampling, and counting the chordal graphs with edge constraints

2010
*
Theoretical Computer Science
*

The objects we look at are

doi:10.1016/j.tcs.2010.03.024
fatcat:5bsf4ic4vraxjibg27k442us54
*chordal**graphs*sandwiched by a given pair*of**graphs*where we assume that at least one*of*the input*graphs*is*chordal*. ... We discuss the problems to*list*, sample, and count the*chordal**graphs*with edge constraints. ... Since the number*of**chordal*completions*of*a*graph*can be quite huge, it would be difficult and even impossible in most*of*the cases to run a*listing*algorithm to obtain the exhaustive*list**of*the*chordal*...##
###
Triangulation Versus Graph Partitioning for Tackling Large Real World Qualitative Spatial Networks

2014
*
2014 IEEE 26th International Conference on Tools with Artificial Intelligence
*

sparse matrix implementation, and (ii)

doi:10.1109/ictai.2014.37
dblp:conf/ictai/Sioutis14
fatcat:stwqqwwnjbb2zetnpd27zlfl2m
*graph**partitioning*and parallelization. ... The proposed techniques for tackling such large networks employ the following two approaches for retaining the sparseness*of*their underlying*graphs*and reasoning with them: (i)*graph*triangulation and ... We would also like to thank Matthias Westphal and Jason Jingshi Li for providing useful feedback on an earlier draft*of*this paper, and for fruitful discussions in general. ...##
###
A clique tree algorithm for partitioning a chordal graph into transitive subgraphs

1995
*
Linear Algebra and its Applications
*

A

doi:10.1016/0024-3795(95)00092-6
fatcat:dgs4rx5jcbfmpkymc2ujuttvfa
*partitioning*problem on*chordal**graphs*that arises in the solution*of*sparse triangular systems*of*equations on parallel computers is considered. ... Several new results about minimal vertex separators in*chordal**graphs*, and in particular, the concept*of*a critical separator*of*a leaf clique, are employed to prove that the new scheme solves the*partitioning*... The*chordal**graph**partitioning*problem is the following: PROBLEM 1. Given a*chordal**graph*G = (V, E), compute a PEO, the associated DAG G d, and an ordered*partition*R1, R 2 ..... ...
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