An Optimal, Edges-Only Fully Dynamic Algorithm for Distance-Hereditary Graphs

The problem of dynamically recognizing a class of graphs has received much attention recently. Given an input graph and a sequence of operations (vertex and edge additions and deletions) to be performed on that graph, the algorithm must determine after each operation if the resulting graph is still a member of the class in question. This paper presents the first dynamic recognition algorithm for distance-hereditary graphs. The algorithm handles edge additions and deletions, and is optimal in that each operation can be performed in constant time. In doing so, the paper completely characterizes when an edge can be added to and removed from a distance-hereditary graph with the result remaining distance-hereditary, and develops a new representation for these graphs in terms of cographs.