Ptolemaic graphs are graphs that satisfy ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. The graph class can also be seen as a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a canonical tree representation based on...

We study on-line colorings of certain graphs given as intersection graphs of objects “between two lines”, i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axisaligned recta...

Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of social networks including common-interest networks, collaboration networks, and actor networks. Simply put, a random intersection graph is constructed by assi...

In this paper we study properties of intersection graphs of kbend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by Bk-VPG. We show here that for every fixed k, Bk-VPG ( Bk+1-VPG and that recognition of graphs from Bk-VPG is NP-complete even when the input graph is given by a Bk...