Rafael B. Teixeira

The polynomial dichotomy for three nonempty part sandwich problems

We classify into polynomial time or NP-complete all three nonempty part sandwich problems. This solves the polynomial dichotomy into polynomial time and NP-complete for this class of graph partition problems.

The sandwich problem for cutsets: clique cutset, k -star cutset

Sandwich problems generalize graph recognition problems with respect to a property Π. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, whose edge set lies between the edge sets of two given graphs. This third graph is required to satisfy a property Π. We present sandwich results corresponding to the polynomial recognition problems: clique cutset, star cutset, and a generalization k-star cutset. We note these graph cutset problems are of interest with respect to sandwich problems. We propose an O(n3)-time polynomial algorithm for star cutset sandwich problem, and an O(n2+k)-time polynomial algorithm for the k-star cutset sandwich problem. We propose an NP-completeness transformation from 1-in-3 3SAT (without negative literals) to clique cutset sandwich problem.

Helly property, clique graphs, complementary graph classes, and sandwich problems

A sandwich problem for property π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property π Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is NP-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.

Skew partition sandwich problem is NP-complete

Abstract Sandwich problems generalize graph recognition problems with respect to a property Π. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, required to satisfy a property Π, whose edge set lies between the edge sets of two given graphs. A skew partition of a graph G = ( V , E ) is a partition of its vertex set V into four nonempty parts A, B, C, D such that each vertex of part A is adjacent to each vertex of part B, and each vertex of part C is nonadjacent to each vertex of part D. Skew cutset generalizes star cutset which in turn generalizes both homogeneous set and clique cutset. Homogeneous set, clique cutset, star cutset, and skew cutset are decompositions arising in perfect graph theory and the recognition of each decomposition is known to be polynomial. Regarding sandwich problems, it is known that homogeneous set sandwich problem is polynomial, clique cutset sandwich problem is NP-complete, and star cutset sandwich problem is polynomial. We prove that skew partition sandwich problem is NP-complete, establishing an interesting computational complexity non-monotonicity.

Helly Property and Sandwich Graphs

The purpose of this paper consists in analysing the recognizing problems of the clique-Helly graphs and the hereditary clique-Helly graphs in sandwich version. Let F be a family of subsets of a set S. We say that F satisfies the Helly property when every subfamily consisting of pairwise intersecting subsets has a non-empty intersection. A graph is clique-Helly when its family of maximal cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. A graph G(V, E) is a sandwich graph for the pair G1(V, E1), G2(V, E2), such that E1 ⊆ E ⊆ E2. A graph sandwich problem consists in, given two graphs G1 e G2, finding a sandwich graph G with a property Π. Graph sandwich problems were defined in the context of Computational Biology and have many applications and it is a natural generalization of recognizing problems [1]. Only problems with polynomial time recognizing are interesting in its sandwich version.

The sandwich problem for cutsets

Abstract A graph G 1 = ( V , E 1 ) is a spanning subgraph of G 2 = ( V , E 2 ) if E 1 ⊆ E 2 ; and a graph G = ( V , E ) is a sandwich graph for the pair G1, G2 if E 1 ⊆ E ⊆ E 2 . A star cutset is a non-empty set C of vertices whose deletion results in a disconnected graph, and such that some vertex in C is adjacent to all the remaining vertices of C. A clique cutset is a vertex cutset which is also a clique. We present an O ( n 3 ) -time algorithm for star cutset sandwich problem ; and we propose an NP-completeness transformation from 1-in-3 3SAT (without negative literals) to clique cutset sandwich problem .

The (k,ℓ) unpartitioned probe problem NP-complete versus polynomial dichotomy

A graph G = ( V , E ) is C probe if V can be partitioned into two sets, probes P and non-probes N, where N is independent and new edges may be added between non-probes such that the resulting graph is in the graph class C . We say that ( N , P ) is a C probe partition for G. The C unpartitioned probe problem consists of an input graph G and the question: Is G a C probe graph? A ( k , ? ) -partition of a graph G is a partition of its vertex set into at most k independent sets and ? cliques. A graph is ( k , ? ) if it has a ( k , ? ) -partition. We prove the full complexity dichotomy into NP-complete and polynomial for ( k , ? ) unpartitioned probe problems: they are NP-complete if k + ? ? 3 , and polynomial otherwise. This gives the first examples of graph classes C that can be recognized in polynomial time but whose probe graph classes are NP-complete. We prove the full complexity dichotomy into NP-complete and polynomial for ( k , ? ) unpartitioned probe problems.The established full complexity dichotomy answers negatively the SPGC conjecture of Le and Ridder, published in WG 2007.The established full complexity dichotomy answers a question of Chang, Hung and Rossmanith, published in DAM 2013.

The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem

The ? graph sandwich problem asks, for a pair of graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) with E 1 ? E 2 , whether there exists a graph G = ( V , E ) that satisfies property ? and E 1 ? E ? E 2 . We consider the property of being F -free, where F is a fixed graph. We show that the claw-free graph sandwich and the bull-free graph sandwich problems are both NP-complete, but the paw-free graph sandwich problem is polynomial. This completes the study of all cases where F has at most four vertices. A skew partition of a graph G is a partition of its vertex set into four nonempty parts A , B , C , D such that each vertex of A is adjacent to each vertex of B , and each vertex of C is nonadjacent to each vertex of D . We prove that the skew partition sandwich problem is NP-complete, establishing a computational complexity non-monotonicity.

The generalized split probe problem

Abstract A generalized split ( k , l ) partition is a vertex set partition into at most k independent sets and l cliques. We prove that the (2, 1) partitioned probe problem is in P whereas the (2, 2) partitioned probe is NP-complete. The full complexity dichotomy into polynomial time and NP-complete for the class of generalized split partitioned probe problems establishes (2, 2) as the first NP-complete self-complementary partitioned probe problem, and answers negatively the PGC conjecture by finding a polynomial time recognition problem whose partitioned probe version is NP-complete.

The (k,ℓ)partitioned probe problem: NP-complete versus polynomial dichotomy

Abstract A graph G = ( V , E ) is C probe if V can be partitioned into two sets, probes P and non-probes N , where N is independent and new edges may be added between non-probes such that the resulting graph is in the graph class C . We say that ( N , P ) is a C probe partition for G . The C partitioned probe problem consists of an input graph G with a vertex partition ( N , P ) and the question: Is ( N , P ) a C probe partition for G ? A ( k , l ) -partition of a graph G is a partition of its vertex set into at most k independent sets and l cliques. A graph is ( k , l ) if it has a ( k , l ) -partition. We prove that the C partitioned probe problem is polynomial for (1,2) and for (2,1)-graphs, and NP-complete for (2,2)-graphs. The results give the first graph class for which the partitioned probe problem is NP-complete, and the full complexity dichotomy into NP-complete and polynomial for ( k , l ) partitioned probe problems: they are NP-complete if k 2 + l 2 ≥ 8 , and polynomial otherwise.