Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the Strong F - closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure , whereas it is a relaxation of F - free Edge Deletion . In Strong F - closure , we want to select a maximum number of edges of the input graph G , and mark them as strong edges , in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F , then the corresponding induced subgraph of G is not isomorphic to F . Hence, the subgraph of G defined by the strong edges is not necessarily F -free, but whenever it contains a copy of F , there are additional edges in G to forbid that strong copy of F in G . We study Strong F - closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F , whereas it does not admit a polynomial kernel even when $$F =P_3$$ F = P 3 . In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d -degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization $$k - \mu (G)$$ k - μ ( G ) , where $$\mu (G)$$ μ ( G ) is the maximum matching size of G . We conclude with some results on the parameterization of Strong F - closure by the number of edges of G that are not selected as strong.

中文翻译：

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强子图闭包的参数化方面

受三元闭包在社交网络中的作用以及找到避免固定模式的最大子图的重要性的启发，我们引入并启动了强 F - 闭包问题的参数化研究，其中 F 是固定图。这是Strong Triadic Closure 的推广，而它是F-free Edge Deletion 的松弛。在强 F 闭包中，我们希望选择输入图 G 的最大边数，并将它们标记为强边，如下所示：每当强边的子集形成与 F 同构的子图时，则对应的G 的诱导子图与 F 不同构。因此，由强边定义的 G 的子图不一定是 F-free 的，但是只要它包含 F 的副本，G 中就有额外的边来禁止 G 中 F 的强副本。我们从具有各种自然参数化的参数化角度研究强 F - 闭包。我们主要关注强边缘的数量 k 作为参数。我们证明了对于每个固定图 F 使用这种参数化的问题是 FPT，而即使当 $$F =P_3$$ F = P 3 时它也不允许多项式核。事实上，后一种情况相当于强三元闭包问题，这促使我们在属于众所周知的图类的输入图上研究这个问题。我们表明，即使输入图是分裂图，强三元闭包也不允许多项式核，而当输入图是平面图时，它允许多项式核，甚至 d 退化。此外，在最大度数最多为 4 的图上，我们证明强三元闭包是具有上述保证参数化的 FPT $$k - \mu (G)$$ k - μ ( G ) ，其中 $$\mu (G)$$ μ ( G ) 是最大匹配大小的 G 。我们总结了一些关于强 F - 闭包参数化的结果，其中 G 的边数未被选为强边。