Intersection Graphs Of Sets Research Articles

Set Labelling Vertices To Ensure Adjacency Coincides With Disjointness

Given a set of nonempty subsets of some universal set, their intersection graph is defined as the graph with one vertex for each set and two vertices are adjacent precisely when their representing sets have non-empty intersection. Sometimes these sets are finite, but in many well known examples like geometric graphs (including interval graphs) they are infinite. One can also study the reverse problem of expressing the vertices of a given graph as distinct sets in such a way that adjacency coincides with intersection of the corresponding sets. The sets are usually required to conform to some template, depending on the problem, to be either a finite set, or some geometric set like intervals, circles, discs, cubes etc. The problem of representing a graph as an intersection graph of sets was first introduced by Erdos [Alon, Noga, Covering graphs by the minimum number of equivalence relations, Combinatorica 6(1986), 201–206] and they looked at minimising the underlying universal set necessary to represent any given graph. In that paper it was shown that the problem is NP complete. In this paper we study a natural variant of this problem which is to consider graphs where vertices represent distinct sets and adjacency coincides with disjointness. Although this is nearly the same problem on the complement graph, for specific families of graphs this is a more natural way of viewing it. The parameter we take into account is the minimum universe size possible (disregarding individual label sizes).

On the interval number of special graphs

AbstractThe interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i(G). Griggs and West showed that $i(G)\le \lceil {1\over 2} (d+1)\rceil $. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally, we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 241–253, 2004

Bounds on the chromatic number of intersection graphs of sets in the plane

In this paper we show that the chromatic number of intersection graphs of congruent geometric figures obtained by translations of a fixed figure in the plane is bounded by the clique number. Further, in the paper we prove that the triangle-free intersection graph of a finite number of compact connected sets with piecewise differentiable Jordan curve boundaries is planar and hence, is 3-colorable.

The interval number of dense graphs

The interval number of a graph G, denoted by i( G), is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t closed intervals. Most known bounds on i( G) are grossly excessive when G has more than half of the possible edges. A plausible remedy is to develop bounds on i( G) that are monotone decreasing in G. Here we bound i( G) in terms of e( G ̄ ) , the number of edges in the complement of G. We prove that i(G)⩽⌈ 1 2 e( G ̄ ) ⌉+ O(n/ log n) .

A sharp edge bound on the interval number of a graph

The interval number of a graph G, denoted by i(G), is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we settle a conjecture of Griggs and West about bounding i(G) in terms of e, that is, the number of edges in G. Namely, it is shown that i(G) ≤ + 1. It is also observed that the edge bound induces i(G) ≤ , where γ(G) is the genus of G. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 153–159, 1999

On the interval number of a chordal graph

AbstractThe interval number of a (simple, undirected) graph G is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t real intervals. A chordal (or triangulated) graph is one with no induced cycles on 4 or more vertices. If G is chordal and has maximum clique size ω(G) = m, then i(G) ⩽ [1 + o(1)]m/log2 m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).

The maximum interval number of graphs with given genus

AbstractThe interval number of a graph G, denoted i(G), is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t compact real intervals. It is known that every planar graph has interval number at most 3 and that this result is best possible. We investigate the maximum value of the interval number for graphs with higher genus and show that the maximum value of the interval number of graphs with genus g is between ⌈√g⌉ and 3 + ⌈√3g⌉. We also show that the maximum arboricity of graphs with genus g is either 1 + ⌈√3g⌉ or 2 + ⌈√3g⌉.

On the chromatic number of multiple interval graphs and overlap graphs

Let χ( G) and ω( G) denote the chromatic number and clique number of a graph G. We prove that χ can be bounded by a function of ω for two well-known relatives of interval graphs. Multiple interval graphs (the intersection graphs of sets which can be written as the union of t closed intervals of a line) satisfy χ⩽2 t( ω−1) for ω⩾2. Overlap graphs satisfy χ⩽2 ω ω 2( ω−1).