Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the ?-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G-S contains no graph from the finite family ? as a minor. It is known that whenever ? contains at least one planar graph, then ?-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size k^{?(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of ?-Minor-Free Deletion for the family ? = {K?, K_{2,3}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with ?(k?) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has ?(k?) vertices and edges

A Turing kernelization dichotomy for structural parameterizations of F-Minor-Free Deletion

For a fixed finite family of graphs F, the F-MINOR-FREE DELETION problem takes as input a graph G and integer ℓ and asks whether a size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-MINOR-FREE DELETION and {K3}-MINOR-FREE DELETION encode VERTEX COVER and FEEDBACK VERTEX SET respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. We show {P3}-MINOR-FREE DELETION parameterized by the feedback vertex number is MK[2]-hard. This rules out the existence of a polynomial kernel assuming NP⊈coNP/poly. Our hardness result generalizes to any F containing only graphs with a connected component of at least 3 vertices, using as parameter the vertex-deletion distance to treewidth min⁡tw(F), where min⁡tw(F) denotes the minimum treewidth of the graphs in F. For all other families F we present a polynomial Turing kernelization. Our results extend to F-SUBGRAPH-FREE DELETION

Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the F-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G- S contains no graph from the finite family F as a minor. It is known that whenever F contains at least one planar graph, then F-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size kO(1) [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of F-Minor-Free Deletion for the family F= { K4, K2,3}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with O(k4) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has O(k4) vertices and edges

A Turing kernelization dichotomy for structural parameterizations of ℱ -minor-free deletion

For a fixed finite family of graphs FF, the FF-Minor-Free Deletion problemtakes as input a graph G and an integer ℓℓ and asks whether there exists a set X⊆V(G)X⊆V(G) of size at most ℓℓ such that G−XG−X is FF-minor-free. For F={K2}F={K2} and F={K3}F={K3} this encodes VertexCover and FeedbackVertex Set respectively. When parameterized by thefeedback vertex number of G these two problems areknown to admit a polynomial kernelization. Such a polynomial kernelization alsoexists for any FF containing a planar graph but no forests. In this paper we show that FF-Minor-Free Deletion parameterizedby the feedback vertex number is MK[2]MK[2]-hard for F={P3}F={P3}. This rules out the existence of a polynomial kernel ssuming NP⊈coNP/polyNP⊈coNP/poly, and also gives evidence that the problem does not admit a polynomialTuring kernel. Our hardness result generalizes to any FF not containing a P3P3-subgraph-free graph, using as parameter the vertex-deletion distance totreewidth mintw(F)mintw(F), where mintw(F)mintw(F) denotes the minimum treewidth of the graphs in FF. For the other case, where FF contains a P3P3-subgraph-free graph, we present a polynomial Turing kernelization. Ourresults extend to FF-Subgraph-Free Deletion