Cláudio Leonardo Lucchesi

Candidate keys for relations

Abstract An algorithm is presented that finds K , the set of all keys for a given set A of attribute names and a given set D [0] of functional dependencies, in time polynomial in | A |, | D [0]| and |K|. It is shown that the problem of deciding whether or not there is a key having cardinality not greater than a specified integer is NP -complete. It is also shown that the problem of deciding whether or not a specified attribute name is prime is NP -complete.

Applications of finite automata representing large vocabularies

The construction of minimal acyclic deterministic partial finite automata to represent large natural language vocabularies is described. Applications of such automata include spelling checkers and advisers, multilanguage dictionaries, thesauri, minimal perfect hashing and text compression.

A pair of forbidden subgraphs and perfect matchings

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to be H-free if no graph in H is an induced subgraph of G. We completely characterize the set H such that every connected H-free graph of sufficiently large even order has a perfect matching in the following cases.(1) Every graph in H is triangle-free. (2) H consists of two graphs (i.e. a pair of forbidden subgraphs).A matching M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H-free graph of sufficiently large odd order has a near-perfect matching in the above cases.

Ear Decompositions of Matching Covered Graphs

G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovász. We also give a simple proof of another theorem due to Lovász which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of or the fourth graph in the sequence is an odd-subdivision of . Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author.

On a Conjecture of Lovász Concerning Bricks

In 1987, Lovasz conjectured that every brick G different from K4, C6, and the Petersen graph has an edge e such that G?e is a matching covered graph with exactly one brick. Lovasz and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. In this paper and its sequel (2001, J. Combin. Theory. Ser. B) we present a proof of this conjecture. We shall in fact prove that if G is a brick different from K4, C6, R8 that does not have the Petersen graph as its underlying simple graph, then it has two edges e and f such that both G?e and G?f are matching covered graphs with exactly one brick, with the additional property that, in each case, the underlying simple graph of that one brick is different from the Petersen graph. A cut C of a matching covered graph G is a separating cut if the two C -contractions of G are matching covered. In this paper, we introduce the notion of the characteristic of a separating cut in a matching covered graph and establish some basic properties. We use those properties to first prove our theorem for solid bricks, that is, bricks which do not have any nontrivial separating cuts. The proof of the theorem for nonsolid bricks will be presented in the sequel.

On clique-complete graphs

Abstract The clique graph, K(G) , of a graph G is the intersection graph of the maximal cliques of G . For a natural number n , a graph G is n-convergent if K n ( G ) is isomorphic to K 1 (the one-vertex graph). A graph G is convergent if it is n-convergent for some natural number n . A 2-convergent graph is called clique - complete . We describe the family of minimal graphs which are clique-complete but have no universal vertices. The minimality used here refers to induced subgraphs. In addition, we show that recognizing clique-complete graphs is Co-NP-complete.

On a Conjecture of Lovász Concerning Bricks: II. Bricks of Finite Characteristic

Abstract In (2001, J. Combin. Theory Ser. B ) we established the validity of the main theorem (1.1) for solid bricks. Here, we establish the existence of suitable separating cuts in nonsolid bricks and prove the theorem by applying induction to cut-contractions with respect to such cuts.

The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in QE belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x ≥ 0 (non-negativity), (ii) x(∂(v)) = 1, for all v ∈ V (degree constraints) and Off) x(∂(S)) ≥ 1, for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.

The Matching Lattice

A set M of edges of a graph G is a matching of G if each vertex of G is incident with at most one edge of M and a perfect matching of G if each vertex of G is incident with precisely one edge of M.

Flow-Critical Graphs

Abstract In this paper we introduce the concept of k-flow-critical graphs. These are graphs that do not admit a k-flow but such that any smaller graph obtained from it by contraction of edges or of pairs of vertices is k-flowable. Any minimal counter-example for Tuttes 3-Flow and 5-Flow Conjectures must be 3-flow-critical and 5-flow-critical, respectively. Thus, any progress towards establishing good characterizations of k-flow-critical graphs can represent progress in the study of these conjectures. We present some interesting properties satisfied by k-flow-critical graphs discovered recently.