Dynamic Distance Hereditary Graphs Using Split Decomposition

The problem of maintaining a representation of a dynamic graph as long as a certain property is satisfied, has recently been considered for a number of properties. This paper presents an optimal algorithm for this problem on vertex-dynamic connected distance hereditary graphs: both vertex insertion and deletion have complexity O(d), where d is the degree of the vertex involved in the modification. Our vertex-dynamic algorithm is competitive with the existing linear time recognition algorithms of distance hereditary graphs, and is also simpler. To achieve this, we revisit the split decomposition by which distance hereditary graphs are known to be completely decomposable. We propose a formulation of this decomposition in terms of graphlabelled trees. Doing so, we are also able to derive an intersection model for distance hereditary graphs, which answers an open problem. ∗Research supported by the French ANR project “Graph Decompositions and Algorithms (GRAAL)” †Research conducted while C. Paul was on Sabbatical at School of Computer Science, McGill University, Montréal, Canada