1. Trivially perfect editing set size?
The question asks if there exists a set F (V x V) of size at most k that makes the graph H = (V, E^F) trivially perfect. Trivially perfect editing is a graph modification problem that has applications in various areas, such as defining community structures in complex networks and related to the well-studied graph parameter tree-depth. Theoretically, some of the earliest NP-Complete problems are graph modification problems, including the Minimum Fill-in problem. Kaplan et al. proved that Minimum Fill-in admits a parameterized algorithm with a kernel containing O(k^2) vertices, which was later improved to O(k^2) vertices by Natanzon et al. Parameterized complexity and kernelization algorithms provide a powerful theoretical framework to cope with decision problems, where a parameterized problem is fixed-parameter tractable if it can be decided in time f(k) * |I| O(1) or has a polynomial kernel.
read more
2. What evidence exists for non-polynomial kernel solutions?
Evidence for non-polynomial kernel solutions exists under certain theoretical complexity assumptions. Specifically, some graph modification problems do not admit polynomial kernels, as shown in references [8, 22, 27, 29]. These findings suggest that certain problems may require more complex approaches beyond polynomial kernels. The Trivially Perfect Editing problem, for example, has been extensively studied and its NP-Completeness was confirmed by Nastos and Gao [30]. However, thanks to Cai's result [7], Trivially Perfect Editing is fixed-parameter tractable, indicating that it can be solved efficiently for certain parameterized cases. Additionally, kernelization algorithms have been developed for this problem, with improvements in the number of vertices required for kernels. Drange and Pilipczuk [14] provided a kernel with O(k^7) vertices, which was later improved to O(k^3) vertices by Dumas et al. [16]. Bathie et al. [1] further improved the bound to O(k^2) vertices for the deletion variant. These advancements in kernelization algorithms demonstrate the ongoing research and development in finding efficient solutions for graph modification problems.
read more
3. What reduction rules are used for trivially perfect graphs?
In the section 'Our contribution', reduction rules are provided for trivially perfect graphs. These rules allow proving the existence of a quadratic vertex-kernel for Trivially Perfect Editing. The reduction rules are divided into two categories: one for reducing trivially perfect modules containing a large matching of non-edges, and another for those without such structures. The latter relies on combs, introduced by Dumas et al. [16], which are composed of shafts and teeth. The reduction rule for shafts ultimately allows bounding the size of trivially perfect modules without large matching of non-edges. The approach also considers unaffected vertices whose neighborhood remains unchanged by an editing of size k. The kernelization algorithm is claimed to work for both deletion and completion variants of the problem.
read more
4. What are the well-known reduction rules essential for obtaining a vertex-kernel?
The well-known reduction rules essential for obtaining a vertex-kernel are Rule 1 and Rule 2. Rule 1 involves removing a connected component from G if G[C] is trivially perfect. Rule 2 involves removing a critical clique from G if |K| > k + 1, removing |K| - (k + 1) arbitrary vertices in K. These rules are safe and can be applied in polynomial time, as stated in Lemma 2 (Folklore, [2, 14]). These reduction rules are crucial in the process of obtaining a vertex-kernel using Theorem 1 [16].
read more