Finding Skew Partitions Efficiently

doi:10.1006/jagm.1999.1122

Journal of Algorithms

Volume 37, Issue 2, November 2000, Pages 505-521

https://doi.org/10.1006/jagm.1999.1122 Get rights and content

Abstract

A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts A, B, C, D such that there are all possible edges between A and B and no edges between C and D. We present a polynomial-time algorithm for testing whether a graph admits a skew partition. Our algorithm solves the more general list skew partition problem, where the input contains, for each vertex, a list containing some of the labels A, B, C, D of the four parts. Our polynomial-time algorithm settles the complexity of the original partition problem proposed by Chvátal in 1985 and answers a recent question of Feder, Hell, Klein, and Motwani.

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Citation Excerpt :
The long-standing challenge was the search for a polynomial-time algorithm or an NP-completeness proof. The first polynomial-time algorithm for testing whether a graph admits a skew partition was obtained in collaboration with Sulamita Klein, Yoshiharu Kohayakawa, and Bruce Reed [26]. The polynomial-time algorithm actually solves the more general list skew partition problem, where the input contains, for each vertex, a list containing some of the four parts.
The Clay Mathematics Institute has selected seven Millennium Problems to motivate research on important classic questions that have resisted solution over the years. Among them is the central problem in theoretical computer science: the P versus NP problem, which aims to classify the possible existence of efficient solutions to combinatorial and optimization problems. The main goal is to determine whether there are questions whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. In this context, it is important to determine precisely what facet of a problem makes it NP-complete. We shall discuss classes of problems for which dichotomy results do exist: every problem in the class is classified into polynomial or NP-complete. 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 classes. More precisely, we have shown that Chvátal’s skew partition is polynomial and that Roberts–Spencer’s clique graph is NP-complete. We have also solved the dichotomy for Golumbic–Kaplan–Shamir’s sandwich problem. We shall describe two examples where we can determine the full dichotomy: the edge-colouring problem for graphs with no cycle with a unique chord and the three nonempty part sandwich problem. Some open problems are discussed: the stubborn problem for list partition, the chromatic index of chordal graphs, and the recognition of split clique graphs.
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We consider the problem of partitioning the vertex-set of a graph into four non-empty sets $A, B, C, D$ such that every vertex of $A$ is adjacent to every vertex of $B$ and every vertex of $C$ is adjacent to every vertex of $D$ . The complexity of deciding if a graph admits such a partition is unknown. We show that it can be solved in polynomial time for several classes of graphs: $K_{4}$ -free graphs, diamond-free graphs, planar graphs, graphs with bounded treewidth, claw-free graphs, $(C_{5}, P_{5})$ -free graphs and graphs with few $P_{4}$ ’s.

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^☆: Research partially supported by CNPq, MCT/FINEP PRONEX Project 107/97, CAPES(Brazil)/COFECUB(France) Project 213/97, FAPERJ, and FAPESP Proc. 96/04505-2.

^f1: [email protected]

^f2: [email protected]

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Regular ArticleFinding Skew Partitions Efficiently☆

Abstract

J. Combin. Theory Ser. B

J. Combin. Theory Ser. B

J. Combin. Theory Ser. B

Discrete Math.

J. Combin. Theory Ser. B

Discrete Math.

J. Algorithms

Discrete Appl. Math.

Regular Article
Finding Skew Partitions Efficiently☆