The paper offers a parallel between two approaches to conceptual clustering, namely formal concept analysis (augmented with the introduction of new operators) and bipartite graph analysis. It is shown that a formal concept (as defined in formal concept analysis) corresponds to the idea of a maximal bi-clique, while a “conceptual world” (defined through a Galois connection associated of the new ...

We formulate the problem of bipartite graph inference as a supervised learning problem, and propose a new method to solve it from the viewpoint of distance metric learning. The method involves the learning of two mappings of the heterogeneous objects to a unified Euclidean space representing the network topology of the bipartite graph, where the graph is easy to infer. The algorithm can be form...

A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the path's sign product is positive or negative. We nd the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph K r;s , which is obtained from the complete bipartite ...

Let F be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An F-free ordering of the vertices of a graph H is a linear ordering of V (H) such that none of the patterns in F occurs as an induced ordered subgraph. We denote by Ord(F) the decision problem asking whether an input graph admits an F-free ordering; we also use Ord(F) to denote the class of graphs that do adm...

Let e(G) denote the number of edges in a graph G, and let Ck denote a k-cycle. It is well-known that every graph has a bipartite subgraph with at least half as many edges. Győri showed that any bipartite, C6-free graph contains a C4-free subgraph containing at least half as many edges. Applying these two results in sequence we see that every C6-free graph, G, has a bipartite C4-free subgraph, H...

We investigate the following generalisation of the ‘multiplication table problem’ of Erdős: given a bipartite graph with m edges, how large is the set of sizes of its induced subgraphs? Erdős’s problem, posed in 1955, of estimating the number of distinct products ab with a, b ≤ n is precisely the problem under consideration when the graph in question is the complete bipartite graph Kn,n. In thi...

A m,n -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D n, a, b consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among m,n bipartite gra...

Popular social networking applications such as Facebook, Twitter, Friendster, etc. generate very large graphs with different characteristics. These social networks are huge, comprising millions of nodes and edges that push existing graph mining algorithms and architectures to their limits. In product-rating graphs, users connect with each other and rate items in tandem. In such bipartite graphs...

For each s ≥ 2, there exists m0 such that the following holds for all m ≥ m0: Let G be a bipartite graph with n = ms vertices in each partition set. If m is odd and minimum degree δ(G) ≥ n+3s 2 − 2, then G contains m vertex-disjoint copies of Ks,s. If m is even, the same holds under the weaker condition δ(G) ≥ n/2+ s− 1. This is sharp and much stronger than a conjecture of Wang [25] (for large n).

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all biparti...