Celina M. Herrera de Figueiredo

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 generation of bicliques of a graph

An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B=X∪Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X,Y≠O, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. When the requirement that X and Y are independent sets of G is dropped, we have a non-induced biclique. We show that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique. We propose an algorithm that generates all non-induced bicliques of a graph. In addition, we propose specialized efficient algorithms for generating the bicliques of special classes of graphs.

Generating bicliques of a graph in lexicographic order

An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B = X ∪ Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X, Y ≠ 0, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. We present an algorithm that generates all bicliques of a graph in lexicographic order, with polynomial-time delay between the output of two successive bicliques. We also show that there is no polynomial-time delay algorithm for generating all bicliques in reverse lexicographic order, unless P = NP. The methods are based on those by Johnson, Papadimitriou and Yannakakis, in the solution of these two problems for independent sets, instead of bicliques.

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvátal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphs

On decision and optimization ( k, l )-graph sandwich problems

A graph G is (k,l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k,l)-Graph Sandwich Problem asks, given two graphs G1 = (V,E1) and G2 = (V,E2), whether there exists a graph G = (V,E) such that E1 ⊆ E ⊆ E2 and G is (k,l). In this paper, we prove that the (k,l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k=l=2. This completely classifies the complexity of the (k,l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l > 2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k ≤ 2 or Δ ≤ 3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant e, such that the existence of an e-approximative algorithm for MIN-(2,1)-GSP implies P = NP.

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.

Optimizing Bull-Free Perfect Graphs

A bull is a graph with five vertices a,b,c,d,e and five edges ab, ac, bc, da, eb. Here we present polynomial-time combinatorial algorithms for the optimal weighted coloring and weighted clique problems in bull-free perfect graphs. The algorithms are based on a structural analysis and decomposition of bull-free perfect graphs.

The stable marriage problem with restricted pairs

A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. In an instance of the STABLE MARRIAGE problem, each of the n men and n women ranks the members of the opposite sex in order of preference. It is well known that at least one stable matching exists for every STABLE MARRIAGE problem instance. We consider extensions of the STABLE MARRIAGE problem obtained by forcing and by forbidding sets of pairs. We present a characterization for the existence of a solution for the STABLE MARRIAGE WITH FORCED AND FORBIDDEN PAIRS problem. In addition, we describe a reduction of the STABLE MARRIAGE WITH FORCED AND FORBIDDEN PAIRS problem to the STABLE MARRIAGE WITH FORBIDDEN PAIRS problem. Finally, we also present algorithms for finding a stable matching, all stable pairs and all stable matchings for this extension. The complexities of the proposed algorithms are the same as the best known algorithms for the unrestricted version of the problem.