Interval graphs and related topics

Interval graphs have engaged the interest of researchers for over twenty-five years. The scope of current research in this area now extends to the mathematical and algorithmic properties of interval graphs, their generalizations, and graph parameters motivated by them. One main reason for this increasing interest is that many real-world applications involve solving problems on graphs which are either interval graphs themselves or are related to interval graphs in a natural way. The problem of characterizing interval graphs was first posed independently by Hajos [45] in combinatorics and by Benzer [17] in genetics. By definition, an interual graph is the intersection graph of a family of intervals on the real line, i.e., to every vertex of the graph there corresponds an interval and two vertices are connected by an edge of the graph if and only if their corresponding intervals have nonempty intersection. However, this suggests a more general paradigm for studying various classes of graphs which can be described as follows. Let9=[&,..., S,] be a family of nonempty subsets of a set S. The subsets are not necessarily distinct. We will call S the host and the subsets Si the objects. In addition, there may or may not be certain constraints placed on the objects such as not allowing one subset to properly contain another or requiring that each subset satisfies a special property. The intersection graph of 9 has vertices Ul,...? II,, with Ui and vi joined by an edge if and only if Si n Sj # 0. We call the pair X = (S, 9) an intersection representation hypergruph for G or more simply a representation. When 9 is a family of intervals on a line, G is an interval graph and X is an interval hypergraph. If we add the constraint that no interval may properly contain another, then we obtain the class of proper interval graphs. When 9 is allowed to be an arbitrary family of sets, the class obtained as intersection graphs is all undirected graphs. On one hand, research has been directed towards intersection graphs of families having some specific topological or other structure. These include circular-arcs, paths in trees, chords of circles, cliques of graphs, and others [35]. On the other hand, certain well-known classes of graphs have subsequently been characterized in terms of intersection graphs. For example, triangulated graphs (every cycle of length 24 has a chord) are the intersection graphs of subtrees of a tree [21, 33,