Dynamic Data Structure for Tree-Depth Decomposition

We present a dynamic data structure for representing a graph G with tree-depth at most D. Tree-depth is an important graph parameter which arose in the study of sparse graph classes. The structure allows addition and removal of edges and vertices such 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, with time bounds for removal depending only on D and constant-time testing of the property, while the time for the initialization and insertion also depends on the size of the formula expressing the property. The concept of tree-depth, introduced in [12], appears prominently in the sparse graph theory and in particular the theory of graph classes with bounded expansion, developed mainly by Nešetřil and Ossona de Mendez [11, 13, 14, 15, 16, 17]. One of its many equivalent definitions is as follows. The tree-depth td(G) of an undirected simple graph G is the smallest integer t for that there exists a rooted forest T of height t with vertex set V (G) such that for every edge xy of G, either x is ancestor of y in T or vice versa—in other words, G is a subgraph of the closure of F . Alternatively, tree-depth can be defined using (and is related to) rank function, vertex ranking number, minimum elimination tree or weak-coloring numbers. Futhermore, a class of graphs closed on subgraphs has bounded tree-depth if and only if it does not contain arbitrarily long paths. Tree-depth is also related to other structural graph parameters—it is greater or equal to path-width (and thus also tree-width), and smaller or equal to the smallest vertex cover. The work leading to this invention has received funding from KONTAKT II LH12095 and SVV 267313.