<p><b>Abstract</b>—Let <tmath>$G$</tmath> be an undirected graph and <tmath>$T$</tmath> be a spanning tree of <tmath>$G$</tmath>. In this paper, an efficient parallel algorithm is proposed for determining whether <tmath>$T$</tmath> is an unordered depth-first search tree of <tmath>$G$</tmath>. The proposed algorithm runs in <tmath>$O(m/p + \log m)$</tmath> time using <tmath>$p$</tmath> processors on the EREW PRAM, where <tmath>$m$</tmath> is the number of edges contained in <tmath>$G$</tmath>. It is cost-optimal and achieves linear speedup.</p>