The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being a... more The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique.
In the paper a characterization of two subclasses of chordal graphs is given: starlike graphs and... more In the paper a characterization of two subclasses of chordal graphs is given: starlike graphs and starlike-threshold graphs. The characterization includes the intersection model (substars of a star), classification by forbidden subgraphs and other structural descriptions.
ABSTRACT A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecti... more ABSTRACT A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecting hyperedges, has a common vertex. We consider the concepts of bipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way as conformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal and bipartite-Helly hypergraphs are also dual. They are useful for characterizing biclique matrices and biclique graphs, that is, the incident biclique-vertex incidence matrix and the intersection graphs of the maximal bicliques of a graph, respectively. These concepts play a similar role for the bicliques of a graph, as do clique matrices and clique graphs, for the cliques of the graph. We describe polynomial time algorithms for recognizing bipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.
The Brazilian Symposium on Graphs, Algorithms and Combinatorics will be held in Fortaleza, Ceará,... more The Brazilian Symposium on Graphs, Algorithms and Combinatorics will be held in Fortaleza, Ceará, Brazil, March 17-19, 2001, in combination with the Brazilian Summer School on Combinatorics and Algorithms, to be held at this same place, March 12-16 and 20-24.Combinatorics has been a very active area, throughout the world, with a strong emphasys in graph theory. The connections with algorithms
ABSTRACT Let G be a graph whose vertices are contaminated. Assigning a searcher to a contaminated... more ABSTRACT Let G be a graph whose vertices are contaminated. Assigning a searcher to a contaminated vertex makes it to become guaraded. Removing the searcher of a guarded vertex turns it clear. However, a clear vertex becomes again contaminated if it has a contaminated neighbour. The node-search number of G is the least number of searchers needed to clear all its vertices. J. Gustedt [Discrete Appl. Math. 45, No. 3, 233-248 (1993; Zbl 0798.68134)] has shown that the problem of determining the node-search number of G is NP-hard for uniform k-starlike graphs. These graphs are generalizations of split graphs, when each vertex of the independent set of its bipartition is replaced by a clique formed by k vertices of identical neighbourhood. We describe necessary and sufficient conditions for finding the node-search number of a uniform k-starlike graph. The characterization described extends a corresponding result for split graphs by T. Kloks [Treewidth. Computations and approximations. Lecture Notes in Computer Science. 842 (Springer-Verlag, Berlin) (1994; Zbl 0825.68144)]. In addition, it leads to a new algorithm for finding the node-search number for graphs of this class.
A graph G is clique-perfect if the cardinality of a maximum clique- independent set of H is equal... more A graph G is clique-perfect if the cardinality of a maximum clique- independent set of H is equal to the cardinality of a minimum clique- transversal of H, for every induced subgraph H of G. When equality holds for every clique subgraph of G, the graph is c{clique-perfect. A graph G is K-perfect when its clique graph K(G) is perfect. In this work, relations are described among the classes of perfect, K- perfect, clique-perfect and c{clique-perfect graphs. Besides, partial characterizations of K-perfect graphs using polyhedral theory and clique subgraphs are formulated.
The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being a... more The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique.
In the paper a characterization of two subclasses of chordal graphs is given: starlike graphs and... more In the paper a characterization of two subclasses of chordal graphs is given: starlike graphs and starlike-threshold graphs. The characterization includes the intersection model (substars of a star), classification by forbidden subgraphs and other structural descriptions.
ABSTRACT A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecti... more ABSTRACT A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecting hyperedges, has a common vertex. We consider the concepts of bipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way as conformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal and bipartite-Helly hypergraphs are also dual. They are useful for characterizing biclique matrices and biclique graphs, that is, the incident biclique-vertex incidence matrix and the intersection graphs of the maximal bicliques of a graph, respectively. These concepts play a similar role for the bicliques of a graph, as do clique matrices and clique graphs, for the cliques of the graph. We describe polynomial time algorithms for recognizing bipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.
The Brazilian Symposium on Graphs, Algorithms and Combinatorics will be held in Fortaleza, Ceará,... more The Brazilian Symposium on Graphs, Algorithms and Combinatorics will be held in Fortaleza, Ceará, Brazil, March 17-19, 2001, in combination with the Brazilian Summer School on Combinatorics and Algorithms, to be held at this same place, March 12-16 and 20-24.Combinatorics has been a very active area, throughout the world, with a strong emphasys in graph theory. The connections with algorithms
ABSTRACT Let G be a graph whose vertices are contaminated. Assigning a searcher to a contaminated... more ABSTRACT Let G be a graph whose vertices are contaminated. Assigning a searcher to a contaminated vertex makes it to become guaraded. Removing the searcher of a guarded vertex turns it clear. However, a clear vertex becomes again contaminated if it has a contaminated neighbour. The node-search number of G is the least number of searchers needed to clear all its vertices. J. Gustedt [Discrete Appl. Math. 45, No. 3, 233-248 (1993; Zbl 0798.68134)] has shown that the problem of determining the node-search number of G is NP-hard for uniform k-starlike graphs. These graphs are generalizations of split graphs, when each vertex of the independent set of its bipartition is replaced by a clique formed by k vertices of identical neighbourhood. We describe necessary and sufficient conditions for finding the node-search number of a uniform k-starlike graph. The characterization described extends a corresponding result for split graphs by T. Kloks [Treewidth. Computations and approximations. Lecture Notes in Computer Science. 842 (Springer-Verlag, Berlin) (1994; Zbl 0825.68144)]. In addition, it leads to a new algorithm for finding the node-search number for graphs of this class.
A graph G is clique-perfect if the cardinality of a maximum clique- independent set of H is equal... more A graph G is clique-perfect if the cardinality of a maximum clique- independent set of H is equal to the cardinality of a minimum clique- transversal of H, for every induced subgraph H of G. When equality holds for every clique subgraph of G, the graph is c{clique-perfect. A graph G is K-perfect when its clique graph K(G) is perfect. In this work, relations are described among the classes of perfect, K- perfect, clique-perfect and c{clique-perfect graphs. Besides, partial characterizations of K-perfect graphs using polyhedral theory and clique subgraphs are formulated.
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Papers by Jayme Szwarcfiter