Sulamita Klein

Complexity of graph partition problems

We introduce a parametrized family of graph problems that includes several well-known graph partition problems as special czses. We develop tools which allow us to classify the complexity of many problems in this family, and in particular lead us to a complete classification for small values of the parameters. Along the way, we obtain a variety of specific results including the following: a generalization of a communication bound on the number of clique-versus-independentset separators; polynomial-time algorithms to recognize generalized split graphs; and, a quasi-polynomial algorithm for the Skew Cutset Problem that essentially resolves an open problem posed by Chv&tal. The last two problems have interesting connections to the Strong Perfect Graph Conjecture of Berge. We also observe that the dichotomy (NPcomplete versus polynomial-time solvable) conjectured for certain graph homomorphism problems, would, if true, imply a slightly weaker dichotomy (NP-complete versus quasipolynomial) for our graph partition problems.

Finding Skew Partitions Efficiently

A skew partition as defined by Chvatal 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 Chvatal in 1985 and answers a recent question of Feder, Hell, Klein, and Motwani.

The homogeneous set sandwich problem

Abstract The graph sandwich problem for property Φ is defined as follows: Given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) such that E 1 ⊆ E 2 , is there a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 which satisfies property Φ? We present a polynomialtime algorithm for solving the graph sandwich problem, when property Φ is “to contain a homogeneous set”. The algorithm presented also provides the graph G and a homogeneous set in G in case it exists.

On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs

Golumbic, Kaplan, and Shamir, in their paper [M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, J. Algorithms 19 (1995) 449-473] on graph sandwich problems published in 1995, left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open. We prove that the sandwich problem for strongly chordal graphs is NP-complete. We also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs.

FINDING H-PARTITIONS EFFICIENTLY ∗

We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external constraints only (no internal constraints), each part non-empty. We describe tools that yield for each problem considered in this paper a simple and low complexity polynomial-time algorithm. Mathematics Subject Classification. 05C85, 68R10.

The graph sandwich problem for 1-join composition is NP-complete

A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets AL, AR , SL and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to vertex of AR∪SR; no vertex of SR is adjacent to a vertex of AL∪SL. The graph sandwich problem for 1-join composition is defined as follows: Given a vertex set V, a forced edge set E1, and a forbidden edge set E3, is there a graph G= (V,E) such that E1 ⊆ E and E ∩ E3 = φ, which is a 1-join composition graph? We prove that the graph sandwich problem for 1-join composition is NP-complete. This result stands in contrast to the case where SL = φ (SR = φ), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution.

Characterizing (k,l)–partitionable Cographs

Abstract We consider the problem of partitioning a graph into k independent sets and l cliques, known as the ( k , l ) -partition problem, which was introduced by Brandstadt in [A. Bransdstadt, Partitions of graphs into one or two independent sets and cliques, Discrete Mathematics 152 (1996) 47–54], and generalized by Feder et al. in [T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition problems, in: Thirty First Annual ACM Symposium on Theory of Computing (1999), Plenum Press, New York, 1999, 464–472] as the M-partition problem. Brandstadt proved that given a graph G, it is NP-complete to decide if G is a ( k , l ) -graph for k ≥ 3 or l ≥ 3 . Since then, a lot of work have been done in order to solve the ( k , l ) -partition problem for many subclasses of perfect graphs. In this work, we consider a subclass of perfect graphs: the cographs, which correspond to graphs without paths with size 4. More precisely, we provide a characterization of cographs that are ( k , l ) -graphs by forbidden configurations, that is, a cograph G is a ( k , l ) -graph if and only if it does not contain any of the forbbiden configurations.

Packing r-Cliques in Weighted Chordal Graphs

In 1, we have previously observed that, in a chordal graph G, the maximum number of independent r-cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr, with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r-cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r-clique in G, we seek a set of independent r-cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r-cliques have the same weight, we simplify the algorithms reported in 1, although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

On generalized split graphs

Abstract Abstract We prove that in a chordal graph the maximum number of independent (i.e., disjoint and nonadjacent) K r s equals the minumum number of cliques that meet all K r s. When r = 1, this implies that chordal graphs are perfect. When r = 2, it contains a well known forbidden subgraph characterization of split graphs. We also discuss algorithms for both these problems. In particular, we illustrate the techniques by giving a new simple recognition algorithm for split graphs. We apply these results to the following generalization of split graphs: A graph is said to be a ( k, l )-graph if its vertex set can be partitioned into k independent sets and l cliques. Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by ( k, l )-graphs in general. (For instance, being a ( k , 0)-graph is equivalent to being k -colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our result gives a forbidden subgraph characterization of (and a polynomial time recognition algorithm for) chordal ( k, l )-graphs.

An algorithm for finding homogeneous pairs

A homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one.