A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple-free graphs, two families where the well-known Lexicographic Breadth First Search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and tha...

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (inter...

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition i...

For two xed graphs X and Y the X Y intersection graph of a graph G is a graph whose vertices are induced subgraphs of G isomor phic to Y and where two vertices are adjacent if their intersection in G contains an induced subgraph isomorphic to X A conformal k graph is a simple hypergraph whose hyperedges are exactly k cliques in the section of the hypergraph Let kX denote the disjoint union of k...

A cycle C of length k is extendable if there is a cycle C′ of length k+1 with V (C) ⊂ V (C′). A graph G = (V,E) of order n is cycle extendable when every cycle C of length k < n is extendable. A chordal graph is a spider intersection graph if it admits an intersection representation which consists of subtrees of a sub-divided star (or spider). In 1990, Hendry conjectured that all hamiltonian ch...

In this report, we present the intersection graphs and describe the role they play in computations of treewidth. We introduce minimal triangulations, minimal separators, potential maximal cliques and blocks as the main tools for exact computation and approximation of that important graph parameter. The first chapter is an introduction to the intersection graph theory. We give all the necessary ...

An intersection representation of a graph G is a function f : V…G† ! 2 (where S is any set) with the property that uv A E…G† if and only if f …u†V f …v†0h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that p...

A k-dimensional box is the cartesian product R1 ×R2 × · · · ×Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a ...

Given a convex polygon of n sides, one can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. We prove that for every convex polygon this graph is pl...

For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. Continuing the study of these matrices associated to an oriented hypergraph, several related structures are investigated including: the incidence dual, the intersection graph (line graph), and the 2-section. The intersection graph is show...