A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs

Algorithms used in data mining and bioinformatics have to deal with huge amount of data efficiently. In many applications, the data are supposed to have explicit or implicit structures. To develop efficient algorithms for such data, we have to propose possible structure models and test if the models are feasible. Hence, it is important to make a compact model for structured data, and enumerate all instances efficiently. There are few graph classes besides trees that can be used for a model. In this paper, we investigate distance-hereditary graphs. This class of graphs consists of isometric graphs and hence contains trees and cographs. First, a canonical and compact tree representation of the class is proposed. The tree representation can be constructed in linear time by using prefix trees. Usually, prefix trees are used to maintain a set of strings. In our algorithm, the prefix trees are used to maintain the neighborhood of vertices, which is a new approach unlike the lexicographically breadth-first search used in other studies. Based on the canonical tree representation, efficient algorithms for the distance-hereditary graphs are proposed, including linear time algorithms for graph recognition and graph isomorphism and an efficient enumeration algorithm. An efficient coding for the tree representation is also presented; it requires 3.59n bits for a distance-hereditary graph of n vertices and 3n bits for a cograph. The results of coding improve previously known upper bounds (both are 2 O(n log n)) of the number of distance-hereditary graphs and cographs to 2 3.59n and 23n , respectively. The extended abstract was presented at the 4th Annual Conference on Theory and Applications of Models of Computation (TAMC07). This work was partially done while the second and third authors were visiting ETH Zürich, Switzerland.