In this paper we propose a simple algorithm called CliqueMinTriang for computing a minimal triangulation of a graph. If F is the set of edges that is added to G to make it a complete graph Kn then the asymptotic complexity of CliqueMinTriang is O(|F|(δ2+|F|)) where δ is the degree of the subgraph of Kn induced by F. Therefore our algorithm performs well when G is a dense graph. We also show how to exploit the existing minimal triangulation techniques in conjunction with CliqueMinTriang to efficiently find a minimal triangulation of nondense graphs. Finally we show how the algorithm can be adapted to perform a backward stepwise selection of decomposable Markov networks; the resulting procedure has the same time complexity as that of existing similar algorithms.