Structured Families of Graphs: Properties, Algorithms, and Representations

The intersection graph of a collection of sets F is the graph obtained by assigning a distinct vertex to each set in F and joining two vertices by an edge precisely when their corresponding sets have a nonempty intersection. When F is allowed to be an arbitrary family of sets, the class of graphs obtained as intersection graphs is simply all undirected graphs. When the types of sets allowed in F is limited, interesting structured families of graphs result. The problem of characterizing the intersection graphs of families of sets having some specific topological or other pattern is often very interesting and frequently has applications to the real world. One of the well-known structured families of intersection graphs is the family of interval graphs. The interval graphs arise when the sets in F are intervals in the real line, that is, a graph G = (V,E) is an interval graph if each vertex v ∈ V can be assigned a real interval Iv so that xy ∈ E ⇔ Ix ∩ Iy 6= ∅. The set of intervals {Iv|v ∈ V } is an interval representation of G. The class of tolerance graphs is a generalization of interval graphs. Tolerance graphs are constructed from intersecting intervals in a manner similar to interval graphs, by putting an edge between two vertices depends on measuring the size of the intersection of their two intervals before declaring that an edge exists. This work was mainly motivated by the study of tolerance graphs, where we have investigated families of graphs that are not necessarily tolerance, but generalize the tolerance model. Our earlier work [26], focused on the interval probe graphs, a subfamily of tolerance graphs, that are used in physical mapping of DNA. The new results reported in the thesis are organized in two major parts. In Part I we introduce the chordal probe graphs, which are a generalization of interval probe graphs, but are not a subfamily of tolerance graphs. In Part II we investigate an approach to generalizing tolerance graphs by replacing the real line by a tree and replacing the role of intervals by either paths or other types of subtree. Part II deals with tolerance models of paths and other types of subtrees. In particular, we study the edge intersection graphs of paths in a tree. In Part I, we introduce the class of chordal probe graphs which are a generalization of both interval probe graphs and chordal graphs. A graph G = (V,E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We show that chordal probe graphs may contain neither an odd-length chordless cycle nor the complement of a chordless cycle, hence they are perfect graphs. We