In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. This problem has been considered in several articles in the last years. We focus on its kernelization , which had been almost unexplored so far. We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly . Motivated by a question of Boria et al. [Discret. Appl. Math. 2015] about the existence of subquadratic kernels for MMVC parameterized by k, we rule out their existence unless P=NP, if we restrict the kernelization algorithms to apply only a type of natural reduction rules that we call "large optimal preserving rules". In particular, these rules contain the typical reduction rules to obtain linear kernels for Vertex Cover . On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a %DIFDELCMD < {\sf %%% %DIFDELCMD < {\sf %%% %DIFDELCMD < {\sf %%% question of Boria et al. [Discret. Appl. Math. 2015] , and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal . On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition . Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless{\sf NP \subseteq{\sf coNP /{\sf poly . This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC .
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This problem has been considered in several articles in the last years. We focus on its kernelization , which had been almost unexplored so far. We prove that MMVC does not admit polynomial kernels parameterized by the size of
Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering
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minimum vertex cover, even on bipartite graphs, unless
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NP
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\subseteq
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coNP
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/
D
poly
D
. Motivated by a
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about the existence of subquadratic kernels for MMVC parameterized by k, we rule out their existence unless P=NP, if we restrict the kernelization algorithms to apply only a type of natural reduction rules that we call "large optimal preserving rules". In particular, these rules contain the typical reduction rules to obtain linear kernels for Vertex Cover
, and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal
A
given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs,
A
. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless
A
NP
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\subseteq
A
coNP
A
/
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poly
A
. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC