Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multi-type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Perio...

Abstract—We consider secure and reliable connectivity in wireless sensor networks that utilize a heterogeneous random key predistribution scheme. We model the unreliability of wireless links by an on-off channel model that induces an Erdős-Rényi graph, while the heterogeneous scheme induces an inhomogeneous random key graph. The overall network can thus be modeled by the intersection of both gr...

We prove that a distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local graph. Then we construct a partial linear space from the local graph to display the contradiction.

We show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs and of distance-hereditary graphs can be solved in O(n4) time. We show that the independent set problem on edge-clique g...

The biclique cover number (resp. biclique partition number) of a graph G, bc(G) (resp. bp(G)), is the least number of bicliques—complete bipartite subgraphs—that are needed to cover (resp. partition) the edges of G. The local biclique cover number (resp. local biclique partition number) of a graph G, lbc(G) (resp. lbp(G)), is the least r such that there is a cover (resp. partition) of the edges...

This paper addresses the problem of coordinating multiple robots travelling through an intersection along fixed paths with positive velocities and kinodynamic constraints. The approach relies on a novel tool: a priority graph that encodes the relative order of the robots at the intersection. The overall planning approach can be decomposed into two key components as follows. The entry of robots ...

Edge-intersection graphs of paths in grids are graphs that can be represented with vertices as paths in grids and edges between the vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was ...

We study the complexity of the colouring problem for circle graphs. We will solve the two open questions of [Un88], where first results were presented. 1. Here we will present an algorithm which solves the 3-colouring problem of circle graphs in time O(n log(n)). In [Un88] we showed that the 4-colouring problem for circle graphs is NP-complete. 2. If the largest clique of a circle graph has siz...

A box graph is the intersection graph of a finite set of orthogonal rectangles in the plane. The problem of whether or not the maximum independent set problem (MIS for short) for box graphs can be approximated within a substantially sub-logarithmic factor in polynomial time has been open for several years. We show that for box graphs on n vertices which have an independent set of size Ω(n/ log ...

Unit-Disk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each circle is specified by the coordinates of its center. Three equivalent graph models can be defined with vertices representing the circles [18]. In the intersection graph model, two vertices are adjacent if the corresponding...