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List Partitions of Chordal Graphs [chapter]

Tomás Feder, Pavol Hell, Sulamita Klein, Loana Tito Nogueira, Fábio Protti
2004 Lecture Notes in Computer Science  
We show that many Mpartition problems that are NP-complete in general become solvable in polynomial time for 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.  ... 
doi:10.1007/978-3-540-24698-5_14 fatcat:fb7vjcua3rcetplb4jqzpgh4xa

List matrix partitions of chordal graphs

Tomás Feder, Pavol Hell, Sulamita Klein, Loana Tito Nogueira, Fábio Protti
2005 Theoretical Computer Science  
Such (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.  ... 
doi:10.1016/j.tcs.2005.09.030 fatcat:qnola5hxpjdb3go5wd2ia5zu44

Polarity of chordal graphs

Tınaz Ekim, Pavol Hell, Juraj Stacho, Dominique de Werra
2008 Discrete Applied Mathematics  
They are defined by the existence 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.  ... 
doi:10.1016/j.dam.2008.01.026 fatcat:qjcpdwgb5bfu3hj6ddraft2fpa

Minimal obstructions for a matrix partition problem in chordal graphs [article]

Juan Carlos García-Altamirano, César Hernández-Cruz
2020 arXiv   pre-print
If M is an m × m matrix over { 0, 1, ∗}, an M-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  ... 
arXiv:2004.01229v1 fatcat:mol265vseffj7p2nalkdqwweom

The P versus NP–complete dichotomy of some challenging problems in graph theory

Celina M.H. de Figueiredo
2012 Discrete Applied Mathematics  
Some open problems are discussed: the stubborn problem for 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.  ... 
doi:10.1016/j.dam.2010.12.014 fatcat:vou3y5m4izgzpcgze6ocw6llrm

Obstructions to partitions of chordal graphs

Tomás Feder, Pavol Hell, Shekoofeh Nekooei Rizi
2013 Discrete Mathematics  
We discuss these problems for the class 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] .  ... 
doi:10.1016/j.disc.2012.05.023 fatcat:hsybgobk7vh57eojttaq472fwm

Star Partitions of Perfect Graphs [chapter]

René van Bevern, Robert Bredereck, Laurent Bulteau, Jiehua Chen, Vincent Froese, Rolf Niedermeier, Gerhard J. Woeginger
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.  ... 
doi:10.1007/978-3-662-43948-7_15 fatcat:hqtybxywcndwhmgnzyybgiqd3q

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

Michel Habib, Ross McConnell, Christophe Paul, Laurent Viennot
2000 Theoretical Computer Science  
By making use 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.  ... 
doi:10.1016/s0304-3975(97)00241-7 fatcat:brlhuthj7zdmfib5ksrfcp6t2m

Partitioning chordal graphs into independent sets and cliques

Pavol Hell, Sulamita Klein, Loana Tito Nogueira, Fábio Protti
2004 Discrete Applied Mathematics  
The algorithm and the characterization extend, in a natural way, to the '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.  ... 
doi:10.1016/s0166-218x(03)00371-8 fatcat:nscz7zaevbakzkl3jp2wdwshby

From a simple elimination ordering to a strong elimination ordering in linear time

J. Sawada, J.P. Spinrad
2003 Information Processing Letters  
We present a linear time algorithm for transforming a simple elimination ordering 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 .  ... 
doi:10.1016/s0020-0190(03)00228-x fatcat:k6wlwyeqfncrfnwynxad3bwwvq

Matrix Partitions of Split Graphs [article]

Tomás Feder, Pavol Hell, Oren Shklarsky
2013 arXiv   pre-print
Matrix 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.  ... 
arXiv:1306.1967v2 fatcat:wbzw246jkffkdp4sj6axkjpiee

Hardness Results and Efficient Algorithms for Graph Powers [chapter]

Van Bang Le, Ngoc Tuy Nguyen
2010 Lecture Notes in Computer Science  
Discrete Math. 18 (2004) 83-102] proved that recognizing squares 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.  ... 
doi:10.1007/978-3-642-11409-0_21 fatcat:eqwta4d65rg43pr6o4grlfyllq

On listing, sampling, and counting the chordal graphs with edge constraints

Shuji Kijima, Masashi Kiyomi, Yoshio Okamoto, Takeaki Uno
2010 Theoretical Computer Science  
The objects we look at are 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  ... 
doi:10.1016/j.tcs.2010.03.024 fatcat:5bsf4ic4vraxjibg27k442us54

Triangulation Versus Graph Partitioning for Tackling Large Real World Qualitative Spatial Networks

Michael Sioutis
2014 2014 IEEE 26th International Conference on Tools with Artificial Intelligence  
sparse matrix implementation, and (ii) 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.  ... 
doi:10.1109/ictai.2014.37 dblp:conf/ictai/Sioutis14 fatcat:stwqqwwnjbb2zetnpd27zlfl2m

A clique tree algorithm for partitioning a chordal graph into transitive subgraphs

Barry W. Peyton, Alex Pothen, Xiaoqing Yuan
1995 Linear Algebra and its Applications  
A 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 .....  ... 
doi:10.1016/0024-3795(95)00092-6 fatcat:dgs4rx5jcbfmpkymc2ujuttvfa
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