arXiv subjects

Marina Groshaus

Publications and source records attributed to Marina Groshaus.

8 recordsLinked to original sources

arXiv2021

Structural properties of biclique graphs and the distance formula

A \textit{biclique} is a maximal induced complete bipartite subgraph of $G$. The \textit{biclique graph} of a graph $G$, denoted by $KB(G)$, is the intersection graph of the family of all bicliques of $G$. In this work we study some structural properties of biclique graphs which are necessary conditions for a graph to be a biclique graph. In particular, we prove that for biclique graphs that are neither a $K_3$ nor a \textit{diamond}, the number of vertices of degree $2$ is less than half the number of vertices in the graph. Also, we present forbidden structures. For this, we introduce a natural definition of the distance between bicliques in a graph. We give a formula that relates the distance between bicliques in a graph $G$ and the distance between their respective vertices in $KB(G)$. Using these results, we can prove not only this new necessary condition involving the degree, but also that some graphs are not biclique graphs. For example, we show that the \textit{crown} is the smallest graph that is not a biclique graph although the known necessary condition for biclique graphs holds, answering an open problem about biclique graphs. Finally, we present some interesting related conjectures and open problems.

cs.DM↗

arXiv2020

Biclique Graphs of $K_3$-free Graphs and Bipartite Graphs

A biclique of a graph is a maximal complete bipartite subgraph. The biclique graph of a graph $G$, $KB(G)$, defined as the intersection graph of the bicliques of $G$, was introduced and characterized in 2010. However, this characterization does not lead to polynomial time recognition algorithms. The time complexity of its recognition problem remains open. There are some works on this problem when restricted to some classes. In this work we give a characterization of the biclique graph of a $K_3$-free graph $G$. We prove that $KB(G)$ is the square graph of a particular graph which we call Mutually Included Biclique Graph of $G$ ($KB_m(G)$). Although it does not lead to a polynomial time recognition algorithm, it gives a new tool to prove properties of biclique graphs (restricted to $K_3$-free graphs) using known properties of square graphs. For instance we generalize a property about induced ${P_3}'$s in biclique graphs to a property about stars and proved a conjecture posted by Groshaus and Montero, when restricted to $K_3$-free graphs. Also we characterize the class of biclique graphs of bipartite graphs. We prove that $KB($bipartite$) = ($IIC-comparability$)^2$, where IIC-comparability is a subclass of comparability graphs that we call Interval Intersection Closed Comparability.

cs.DM↗

arXiv2019

Intersection graph of maximal stars

A biclique of a graph $G$ is an induced complete bipartite subgraph of $G$ such that neither part is empty. A star is a biclique of $G$ such that one part has exactly one vertex. The star graph of $G$ is the intersection graph of the maximal stars of $G$. A graph $H$ is star-critical if its star graph is different from the star graph of any of its proper induced subgraphs. We begin by presenting a bound on the size of star-critical pre-images by a quadratic function on the number of vertices of the star graph, then proceed to describe a Krausz-type characterization for this graph class; we combine these results to show membership of the recognition problem in \textsf{NP}. We also present some properties of star graphs. In particular, we show that they are biconnected, that every edge belongs to at least one triangle, characterize the structures the pre-image must have in order to generate degree two vertices, and bound the diameter of the star graph with respect to the diameter of its pre-image. Finally, we prove a monotonicity theorem, which we apply to list every star graph on at most eight vertices.

math.CO↗

arXiv2016

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

A \emph{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, bounds on the maximum number of bicliques were given. In this paper we study bounds on the minimun 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 for a subclass of $\{C_4$,false-twin$\}$-free graphs and for the class of $\{K_3$,false-twin$\}$-free graphs. Finally we discuss the problem for general graphs.

cs.DM↗

arXiv2015

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 $KB(G)$, is the intersection graph of all the bicliques of $G$. The biclique graph can be thought as an operator between graphs. The iterated biclique graph of $G$ denoted by $KB^{k}(G)$, 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.

cs.DM↗

arXiv2014

Proper Hamiltonian Paths in Edge-Coloured Multigraphs

Given a $c$-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainbow degree and the connectivity.

cs.DM↗

arXiv2013

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.

math.CO↗

arXiv2012

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) L-coloring 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. Specifically, we prove that the star (biclique) k-coloring and k-choosability problems are Σ_2^p-complete and Π_3^p-complete for k > 2, respectively, even when the input graph contains no induced C_4 or K_{k+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, threshold graphs, and net-free block graphs.

cs.DM↗