# Faster and Enhanced Inclusion-Minimal Cograph Completion

We design two incremental algorithms for computing an inclusion-minimal completion of an arbitrary graph into a cograph. The first one is able to do so while providing an additional property which is crucial in practice to obtain inclusion-minimal completions using as few edges as possible : it is able to compute a minimum-cardinality completion of the neighbourhood of the new vertex introduced at each incremental step. It runs in O(n+m') time, where m' is the number of edges in the completed graph. This matches the complexity of the algorithm in [Lokshtanov, Mancini and Papadopoulos 2010] and positively answers one of their open questions. Our second algorithm improves the complexity of inclusion-minimal completion to O(n+mlog^2 n) when the additional property above is not required. Moreover, we prove that many very sparse graphs, having only O(n) edges, require Ω(n^2) edges in any of their cograph completions. For these graphs, which include many of those encountered in applications, the improvement we obtain on the complexity scales as O(n/log^2 n).

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