Canonical Tree Representation of Distance Hereditary Graphs with Applications∗

The class of distance hereditary graphs consists of the isometric graphs. That is, for each pair of vertices, its distance is invariant for any induced path in a distance hereditary graph. In the paper, a canonical tree representation of a distance hereditary graph is proposed. A linear time algorithm for computing the tree representation is also presented. Hence the recognition problem and the graph isomorphism problem for the graph class can be solved in linear time, thanks to linear time isomorphism algorithm for labeled trees. The tree representation takes O(|V |) space for a distance hereditary graph G = (V,E). Hence it can be used as a compact data structure for the graph. It is so informative that all pruning sequences, which is a previously known characterization based on a vertex ordering, can be generated from the tree representation efficiently.