A graph is ptolemaic if and only if it is both chordal and distancehereditary. Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary. The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G). ...

Every chordal graph G admits a representation as the intersection graph of a family of subtrees of a tree. A classic way of finding such an intersection model is to look for a maximum spanning tree of the valuated clique graph of G. Similar techniques have been applied to find intersection models of chordal graph subclasses as interval graphs and path graphs. In this work, we extend those metho...

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the X-axis, distance 1 + (0 < < 1) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unk...

The considered problem concerns constructing independent sets in a random intersection graph. We concentrate on two cases of the model: a binomial and a uniform random intersection graph. For both models we analyse two greedy algorithms and prove that they find asymptotically almost optimal independent sets. We provide detailed analysis of the presented algorithms and give tight bounds on the i...

A clique in a graph is a complete subgraph maximal under inclusion. The clique graph of a graph is the intersection graph of its cliques. A graph is self-clique when it is isomorphic to its clique graph. A circular-arc graph is the intersection graph of a family of arcs of a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. In th...

Two characterizations of hypercubes are given: 1) A graph is a hypercube if and only if it is antipodal and bipartite (0, 2)-graph. 2) A graph is an nhypercube if and only if there are n pairs of prime convexes, the graph is a prime convex intersection graph, and each intersection of n prime convexes (no one of which is from the same pair) is a vertex.

1 Stable 2-Pairs and (X; Y)-Intersection Graphs 2 Abstract Given two xed graphs X and Y , the (X; Y)-intersection graph of a graph G is a graph where 1. each vertex corresponds to a distinct induced subgraph in G that is iso-morphic to Y , and 2. two vertices are adjacent ii the intersection of their corresponding sub-graphs contains an induced subgraph isomorphic to X. This notion generalizes ...