Minimal comparability completions

We study the problem of adding edges to a given arbitrary graph so that the resulting graph is a comparability graph, called a comparability completion of the input graph. Computing a comparability completion with the minimum possible number of added edges is an NP-hard problem. Our purpose here is to add an inclusion minimal set of edges to obtain a minimal comparability completion, which means that no proper subset of the added edges is sufficient to create a comparability completion. We show that this problem is solvable in polynomial time. Minimal completions of arbitrary graphs into chordal graphs have been studied extensively, and new results have been added continuously. There has been an increasing interest in minimal completion problems, and minimal completions of arbitrary graphs into interval graphs and into split graphs have been studied recently. We extend these previous results to comparability graphs, and we give a polynomial time algorithm for computing a minimal comparability completion of an arbitrary input graph. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.