Testing Depth First Search Numbering

Property Testing is a formal framework to study the computational power and complexity of sampling from combinatorial objects. A central goal in standard graph property testing is to understand which graph properties are testable with sublinear query complexity. Here, a graph property P is testable with a sublinear query complexity if there is an algorithm that makes a sublinear number of queries to the input graph and accepts with probability at least 2/3, if the graph has property P, and rejects with probability at least 2/3 if it is $\varepsilon$-far from every graph that has property P. In this paper, we introduce a new variant of the bounded degree graph model. In this variant, in addition to the standard representation of a bounded degree graph, we assume that every vertex $v$ has a unique label num$(v)$ from $\{1, \dots, |V|\}$, and in addition to the standard queries in the bounded degree graph model, we also allow a property testing algorithm to query for the label of a vertex (but not for a vertex with a given label). Our new model is motivated by certain graph processes such as a DFS traversal, which assign consecutive numbers (labels) to the vertices of the graph. We want to study which of these numberings can be tested in sublinear time. As a first step in understanding such a model, we develop a \emph{property testing algorithm for discovery times of a DFS traversal} with query complexity $O(n^{1/3}/\varepsilon)$ and for constant $\varepsilon>0$ we give a matching lower bound.