Summary: We consider the dynamic recognition problem for the class of \(P_{4}\)-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a \(P_{4}\)-sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish conditions for each modification to yield a \(P_{4}\)-sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in \(O(1)\) time and vertex modifications in \(O(d)\) time for a vertex of degree \(d\). Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for \(P_{4}\)-sparse graphs. Moreover, by maintaining the children of each node of the modular decomposition tree in a binomial heap, we can handle vertex deletions in \(O(\log n)\) time, at the expense of needing \(O(\log n)\) time for each edge modification and \(O(d \log n)\) time for the addition of a vertex adjacent to \(d\) vertices.
For the entire collection see [Zbl 1137.68005].