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

Distance-hereditary graphs consist of the isometric graphs, and hence contain trees and cographs. First, a canonical and compact tree representation of the class is proposed. The tree representation can be constructed in linear time using two prefix trees. Usually, the prefix trees are used to maintain a set of strings. The prefix trees in our algorithm are used to maintain the neighbors for each vertex, which is new approach comparing to the other known results based on the lexicographically bread first search. Several efficient algorithms for the distancehereditary graphs based on the canonical tree representation are proposed; linear time algorithms for graph recognition and graph isomorphism, and efficient enumeration algorithm. An efficient coding for the tree representation is also presented, which requires 4n bits for a distance-hereditary graph of n vertices, and 3n bits for a cograph. The results improve previously known upper bounds of the number of distance-hereditary graphs and cographs.

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