Let \(G\) be a graph with even order \(p\) and let \(k\) be a positive integer with \(p \geq 2k + 2\). It is proved that if the toughness of \(G\) is at least \(k\), then the subgraph of \(G\) obtained by deleting any \(2k – 1\) edges or \(k\) vertices has a perfect matching. Furthermore, we show that the results in this paper are best possible.