Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1 + ) n edges (for a prespecified constant 0"> 0 ), where the decision for different edges should be consistent with the same subgraph G . Can this task be performed by inspecting only a {\it constant} number of edges in G ?

Our main results are:

(1) We show that if every t -vertex subgraph of G has expansion 1 ( log t ) 1+ o (1) then one can (deterministically) construct a sparse spanning subgraph G of G using few s inspections. To this end we analyze a ``local'' version of a famous minimum-weight spanning tree algorithm.

(2) We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3 -regular graphs of high girth, in which every t -vertex subgraph has expansion 1 ( log t ) 1 − o (1) .