A dynamic data structure for MSO properties in graphs with bounded tree-depth

Abstract

Tree-depth is an important graph parameter which arose in the study of sparse graph classes. We present a dynamic data structure for representing a graph G with tree-depth at most D. The structure allows addition and removal of edges and vertices under assumption that the resulting graph still has tree-depth at most D, in time bounds depending only on D. A tree-depth decomposition of the graph is maintained explicitly. This makes the data structure useful for dynamization of static algorithms for graphs with bounded tree-depth. As an example application, we give a dynamic data structure for MSO property testing. © 2014 Springer-Verlag Berlin Heidelberg.