A graph has hyuiciry k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes in a &-dimensional space (where the sides of the boxes are parallel to the coordinate axis). A graph has grid dimension k if k is the smallest integer such that G is an intersection graph of k-dimensional boxes (parallel to the coordinate axis) in a (k+ I)-dimensional space. We prove ...

For undirected graphs G (V, E) and G0 (V0 , E0), say that G is a region intersection graph over G0 if there is a family of connected subsets {Ru ⊆ V0 : u ∈ V} of G0 such that {u , v} ∈ E ⇐⇒ Ru ∩ Rv , ∅. We show if G0 excludes the complete graph Kh as a minor for some h > 1, then every region intersection graph G over G0 with m edges has a balanced separator with at most ch √ m nodes, where ch i...

We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a con...

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2 log6 n) time for n-vertex graphs with maximum degree at most three. key words: 3-regular graph, bipartite permutat...

We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we construct OP (log log n) upper bound for the length of the shortest path connecting them. key words: intersection graph, random graph, power law, giant compon...

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the nonnegative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Came...

Constructions and bounds are given for the largest clique that can be represented as an intersection graph of intervals on the real line with bounded depth and multiplicity. Also, for the graph defined on the subsets of a set by putting vertices adjacent if and only if the corresponding subsets intersect, the interval number and boxicity are examined. Ke ywords: intersection graph, interval num...

A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm present...

A Steiner triple system of order n is a collection of subsets of size three, taken from the n-element set {0, 1, ..., n−1}, such that every pair is contained in exactly one of the subsets. The subsets are called triples, and a block-intersection graph is constructed by having each triple correspond to a vertex. If two triples have a non-empty intersection, an edge is inserted between their vert...

In this paper, we show that many graph search problems can be solved quite efficiently for a geometric intersection graph ofhorizontal and vertical line segments. We first extract several basic operations for depth first search and breadth first search on a graph. Then we present data structures for the intersection graph in terms of which those operations can be implemented in an efficient man...