Marina Groshaus

On the Iterated Biclique Operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G, denoted by , is the intersection graph of the bicliques of G. We say that a graph G diverges (or converges or is periodic) under an operator F whenever ( for some m, or for some k and , respectively). Given a graph G, the iterated biclique graph of G, denoted by , is the graph obtained by applying the biclique operator k successive times to G. In this article, we study the iterated biclique graph of G. In particular, we classify the different behaviors of when the number of iterations k grows to infinity. That is, we prove that a graph either diverges or converges under the biclique operator. We give a forbidden structure characterization of convergent graphs, which yield a polynomial time algorithm to decide if a given graph diverges or converges. This is in sharp contrast with the situsation for the better known clique operator, where it is not even known if the corresponding problem is decidable.

Biclique-Helly Graphs

A graph is biclique-Helly when its family of (maximal) bicliques is a Helly family. We describe characterizations for biclique-Helly graphs, leading to polynomial time recognition algorithms. In addition, we relate biclique-Helly graphs to the classes of clique-Helly, disk-Helly and neighborhood-Helly graphs.

The number of convergent graphs under the biclique operator with no twin vertices is finite

Abstract The biclique graph of G, K B ( G ) , is the intersection graph of the bicliques of G. Given a graph G, the iterated biclique graph of G, K B k ( G ) , is the graph defined iteratively as follows: K B k + 1 ( G ) = K B ( K B k ( G ) ) . Say that a graph G diverges (resp. converges) under the operator KB whenever lim k → ∞ V ( K B k ( G ) ) = ∞ (resp. lim k → ∞ K B k ( G ) = K B m ( G ) for some m). Each of these behaviours were recently characterized. These characterizations lead to a O ( n 4 ) time algorithm for deciding the divergence or convergence of a graph. In this work we prove that any graph with at least 7 bicliques diverges under the biclique operator. Furthermore, we prove that graphs with no twin vertices that are not divergent have at most 12 vertices, which leads to a linear time algorithm to decide if a graph converges or diverges under the biclique operator.

On a conjecture concerning helly circle graphs

Dizemos que G e um grafo e-circular se existe uma bijecao entre seus vertices e retas no plano cartesiano de forma que dois vertices sao adjacentes em G se e somente se as retas correspondentes se intersectam dentro do circulo de raio unitario centrado na origem. Esta definicao sugere um metodo para decidir se um dado grafo G e um grafo e-circular, construindo convenientemente um sistema S de equacoes e inequacoes que representa a estrutura de G, de tal modo que G e um grafo e-circular se e somente se S tem solucao. Em realidade, grafos e-circulares sao exatamente os grafos circulares (grafos de intersecao de cordas em um circulo), e portanto este metodo fornece um modo analitico de reconhecimento de grafos circulares. Um grafo G e circular Helly se G e um grafo circular e existe um modelo de cordas de G tal que em todo grupo de tres cordas mutuamente intersectantes existe um ponto comum a todas elas. Uma conjectura de Duran (2000) afirma que G e um grafo circular Helly se e somente se G e um grafo circular e nao contem diamantes induzidos (um diamante e um grafo formado por quatro vertices e cinco arestas). Muitas tentativas infrutiferas - baseadas principalmente em abordagens combinatorias e geometricas - foram realizadas para tentar validar a conjectura. Neste trabalho, utilizamos as ideias subjacentes a definicao de grafos e-circulares e reformulamos a conjectura em termos de uma equivalencia entre dois sistemas de equacoes e inequacoes, fornecendo uma nova ferramenta analitica para trata-la.

The Star and Biclique Coloring and Choosability Problems

A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic. If G admits a star (biclique) Lcoloring for every k-list assignment L, then G is said to be star (biclique) k-choosable. In this article we study the computational complexity of the star and biclique coloring and choosability problems. Specically, we prove that the star (biclique) k-coloring and k-choosability problems are p 2-complete and p 3-complete for k > 2, respectively, even when the input graph contains no induced C4 or Kk+2. Then, we study all these problems in some related classes of graphs, including H-free graphs for every H on three vertices, graphs with restricted diamonds, split graphs, and threshold graphs.

Almost every graph is divergent under the biclique operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G . The biclique graph of G denoted by K B ( G ) , is the intersection graph of all the bicliques of G . The biclique graph can be thought as an operator between the class of all graphs. The iterated biclique graph of G denoted by K B k ( G ) , is the graph obtained by applying the biclique operator k successive times to G . The associated problem is deciding whether an input graph converges, diverges or is periodic under the biclique operator when k grows to infinity. All possible behaviors were characterized recently and an O ( n 4 ) algorithm for deciding the behavior of any graph under the biclique operator was also given. In this work we prove new structural results of biclique graphs. In particular, we prove that every false-twin-free graph with at least 13 vertices is divergent. These results lead to a linear time algorithm to solve the same problem.

Tight lower bounds on the number of bicliques in false-twin-free graphs

A biclique is a maximal bipartite complete induced subgraph of G. Bicliques have been studied in the last years motivated by the large number of applications. In particular, enumeration of the maximal bicliques has been of interest in data analysis. Associated with this issue, upper and lower bounds on the maximun number of bicliques were given. In this paper we study lower bounds on the number of bicliques of a graph. Since adding false-twin vertices to G does not change the number of bicliques, we restrict to false-twin-free graphs. We give a tight lower bound on the minimum number bicliques in {C4,diamond,false-twin}-free graphs, {K3,false-twin}-free graphs and we present some conjectures for general false-twin-free graphs.

On edge-sets of bicliques in graphs

A biclique is a maximal induced complete bipartite subgraph of a graph. We investigate the intersection structure of edge-sets of bicliques in a graph. Specifically, we study the associated edge-biclique hypergraph whose hyperedges are precisely the edge-sets of all bicliques. We characterize graphs whose edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its 2-section) by means of a single forbidden induced obstruction, the triangular prism. Using this result, we characterize graphs whose edge-biclique hypergraph is Helly and provide a polynomial time recognition algorithm. We further study a hereditary version of this property and show that it also admits polynomial time recognition, and, in fact, is characterized by a finite set of forbidden induced subgraphs. We conclude by describing some interesting properties of the 2-section graph of the edge-biclique hypergraph.

Proper Hamiltonian Paths in Edge-Colored Multigraphs

Abstract A c -edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper hamiltonian path is a path containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper hamiltonian path, depending on several parameters such as the number of edges, the rainbow degree, etc.

On star and biclique edge‐colorings

A biclique of G is a maximal set of vertices that induces a complete bipartite subgraph Kp,q of G with at least one edge, and a star of a graph G is a maximal set of vertices that induces a complete bipartite graph K1,q. A biclique (resp. star) edge-coloring is a coloring of the edges of a graph with no monochromatic bicliques (resp. stars). We prove that the problem of determining whether a graph G has a biclique (resp. star) edgecoloring using two colors is NP-hard. Furthermore, we describe polynomial time algorithms for the problem in restricted classes: K3-free graphs, chordal bipartite graphs, powers of paths, and powers of cycles.