Abstract 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 et al., 2010) and positively answers one of their open questions. Our second algorithm improves the complexity of inclusion-minimal completion to O ( n + m log 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 ) .

中文翻译：

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更快和增强的包含 - 最小cograph完成

摘要 我们设计了两种增量算法来计算任意图到 cograph 的包含最小完成。第一个能够这样做，同时提供一个额外的属性，这在实践中使用尽可能少的边获得包含最小完成至关重要：它能够计算引入的新顶点的邻域的最小基数完成每个增量步骤。它在 O ( n + m ′ ) 时间内运行，其中 m ′ 是完整图中的边数。这与 (Lokshtanov et al., 2010) 中算法的复杂性相匹配，并且肯定地回答了他们的一个开放性问题。当不需要上述附加属性时，我们的第二个算法将包含最小完成的复杂性提高到 O ( n + m log 2 n )。此外，我们证明了许多非常稀疏的图，只有 O ( n ) 条边，在它们的任何 cograph 完成中都需要 Ω ( n 2 ) 条边。对于这些图，其中包括许多在应用程序中遇到的图，我们在复杂性尺度上获得的改进为 O ( n ∕ log 2 n ) 。