700 research outputs found
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
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
- …