A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists an edgeweighted tree T and two non-negative real numbers dmin and dmax such that each leaf u of T is a node of V 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 weights of the edges on the unique path from u to v in T . The main issue on these graphs consists in charac...

String graphs, that is, intersection graphs of curves in the plane, have been studied since the 1960s. We provide an expository presentation of several results, including very recent ones: some string graphs require an exponential number of crossings in every string representation; exponential number is always sufficient; string graphs have small separators; and the current best bound on the cr...

We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (Gn,m,~ p) which includes the model of [10] (the “uniform” random intersection graphs model) as an important special case. We also define an interesting variation of the model of...

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...

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have onl...

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a $G$, $\dim_{COG}(G)$ (resp. $\dim_{TH}(G)$) denotes minimum number cographs threshold graphs) whose gives $G$. We present several new bounds on these parameters for general as well some special classes graphs. It shown that any ...

Hartman I.B.-A., I. Newman and R. Ziv, On grid intersection graphs, Discrete Mathematics 87 (1991) 41-52. A bipartite graph G = (X, Y; E) has a grid representation if X and Y correspond to sets of horizontal and vertical segments in the plane, respectively, such that (xi, y,) E E if and only if segments x, and y, intersect. We prove that all planar bipartite graphs have a grid representation, a...

For intersection graphs of disks and other fat objects, polynomial-time approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unit-size objects. We present approximation algorithms and inapproximability results that shed new light on the approximability of...

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k ≥ 0, the intersection graph Qk(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices Hx and Hy in Qk(G) are adjacent whenever the intersection Hx ∩ Hy contains a subgraph isomorphic to Qk. Characterizations of clique...

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, k ≥ 0. If the number k of bends is not restricted, then the VPG graphs are shown to be equivalent to the well-known class of string graphs, namely, the in...