A Fully Dynamic Algorithm for the Recognition of P4-Sparse Graphs

In this paper, we solve the dynamic recognition problem for the class of P4-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognizeif each such modification yields a P4-sparse graph, and if yes, to update a representationof the graph. Our approach relies on maintaining the modular decomposition tree of thegraph, which we use for solving the recognition problem. We establish properties for eachmodification to yield a P4-sparse graph and obtain a fully dynamic recognition algorithmwhich handles edge modifications in O(1) time and vertex modifications in O(d) time for avertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithmand a new optimal incremental algorithm for P4-sparse graphs. Moreover, by maintainingthe children of each node of the modular decomposition tree in a binomial heap, we canhandle vertex deletions in O(logn) time, at the expense of needing O(logn) time for eachedge modification and O(d log n) time for the addition of a vertex adjacent to d vertices.