Several parameters of generalized Mycielskians

The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m?0, one can transform G into a new graph μ_m(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μ_m(G).     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

On the generation of bicliques of a graph

An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B=X∪Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X,Y≠∅, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. When the requirement that X and Y are independent sets of G is dropped, we have a non-induced biclique. We show that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique. We propose an algorithm that generates all non-induced bicliques of a graph. In addition, we propose specialized efficient algorithms for generating the bicliques of special classes of graphs.     

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m²/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.     

Biclique completion problems for multicast network design

This paper considers the problem of aggregating several multicast sessions. A multicast session is defined as a subset of clients requiring the same information. Besides, each client can require several multicast sessions. A telecommunication network cannot manage many multicast sessions at the same time. It is hence necessary to group the sessions into a limited number of clusters. The problem then consists in aggregating the sessions into clusters to limit the number of unnecessary information sent to clients. The strong relationship of the problems with biclique problems in bipartite graph is established. We then model the problems using integer quadratic and linear programming formulations. We investigate some properties to strengthen the models. Several algorithms are provided and compared with a series of numerical experiments.     

On decompositions of complete hypergraphs

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.     

Chromatic characterization of biclique covers

A biclique B of a simple graph G is the edge-set of a complete bipartite subgraph of G. A biclique cover of G is a collection of bicliques covering the edge-set of G. Given a graph G, we will study the following problem: find the minimum number of bicliques which cover the edge-set of G. This problem will be called the minimum biclique cover problem (MBC). First, we will define the families of independent and dependent sets of the edge-set E(G) of G: F⊆E(G) will be called independent if there exists a biclique B⊆E(G) such that F⊆B, and will be called dependent otherwise. From our study of minimal dependent sets we will derive a 0-1 linear programming formulation of the following problem: find the maximum weighted biclique in a graph. This formulation may have an exponential number of constraints with respect to the number of nodes of G but we will prove that the continuous relaxation of this integer program can be solved in polynomial time. Finally we will also study continuous relaxation methods for the problem (MBC). This research was motivated by an open problem of Fishburn and Hammer.