Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n 2 +αm) time and O(n+m) space. (Here n, m, and α = O(m1/2 ) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n + m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n 2 + αm) time and O(αm) space.Fil: Eguía, Martiniano. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Soulignac, Francisco Juan. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On hereditary Helly classes of graphs

Graphs and Algorithm

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.Comment: This version corrects an error in Theorem 11 found after the paper went into prin

Exploring Agreeability in Tree Societies

Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection properties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty intersection. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that the agreement number of (2, m)-agreeable tree societies is at least 1 |S | and 3 that the agreement number of (k, k + 1)-agreeable tree societies is at least |S|−1

Intersection of Longest Paths in Graph Theory and Predicting Performance in Facial Recognition

A set of subsets is said to have the Helly property if the condition that each pair of subsets has a non-empty intersection implies that the intersection of all subsets has a non-empty intersection. In 1966, Gallai noticed that the set of all longest paths of a connected graph is pairwise intersecting and asked if the set had the Helly property. While it is not true in general, a number of classes of graphs have been shown to have the property. In this dissertation, we show that K4-minor-free graphs, interval graphs, circular arc graphs, and the intersection graphs of spider graphs are classes that have this property. The accuracy of facial recognition algorithms on images taken in controlled conditions has improved significantly over the last two decades. As the focus is turning to more unconstrained or relaxed conditions and toward videos, there is a need to better understand what factors influence performance. If these factors were better understood, it would be easier to predict how well an algorithm will perform when new conditions are introduced. Previous studies have studied the effect of various factors on the verification rate (VR), but less attention has been paid to the false accept rate (FAR). In this dissertation, we study the effect various factors have on the FAR as well as the correlation between marginal FAR and VR. Using these relationships, we propose two models to predict marginal VR and demonstrate that the models predict better than using the previous global VR

Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

Graphs and Algorithm