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Genome rearrangements are events where large blocks of DNA exchange places during evolution. The analysis of these events is a promising tool for understanding evolutionary genomics, providing data for phylogenetic reconstruction based on genome rearrangement measures. Many pairwise rearrangement distances have been proposed, based on finding the minimum number of rearrangement events to transform one genome into the other, using some predefined operation. When more than two genomes are considered, we have the more challenging problem of rearrangement-based phylogeny reconstruction. Given a set of genomes and a distance notion, there are at least two natural ways to define the "target" genome. On the one hand, finding a genome that minimizes the sum of the distances from this to any other, called the median genome. On the other hand, finding a genome that minimizes the maximum distance to any other, called the closest genome. Considering genomes as permutations of distinct integers, some distance metrics have been extensively studied. We investigate the median and closest problems on permutations over the following metrics: breakpoint distance, swap distance, block-interchange distance, short-block-move distance, and transposition distance. In biological applications some values are usually very small, such as the solution value d or the number k of input permutations. For each of these metrics and parameters d or k, we analyze the closest and the median problems from the viewpoint of parameterized complexity. We obtain the following results: NP-hardness for finding the median/closest permutation regarding some metrics of distance, even for only k = 3 permutations; Polynomial kernels for the problems of finding the median permutation of all studied metrics, considering the target distance d as parameter; NP-hardness result for finding the closest permutation by short-block-moves; FPT algorithms and infeasibility of polynomial kernels for finding the closest permutation for some metrics when parameterized by the target distance d. [ABSTRACT FROM AUTHOR]

For a fixed finite collection of graphs F, the F-M-DELETION problem is as follows: given an n-vertex input graph G, find the minimum number of vertices that intersect all minor models in G of the graphs in F. by Courcelle's Theorem, this problem can be solved in time fF(tw) - n°(1), where tw is the treewidth of G for some function ƒ depending on F. In a recent series of articles, we have initiated the program of optimizing asymptotically the function ff. Here we provide an algorithm showing that fF(tw) = 2° (tw.log tw) for every collection F. Prior to this work, the best known function ƒ was double- exponential in tw. In particular, our algorithm vastly extends the results of Jansen, Lokshtanov, and Saurabh [Proc. of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2014, pp. 1802-1811] for the particular case F = {K5, K3,3} and of Kociumaka and Pilipczuk [Algorithmica, 81 (2019), pp. 3655-3691] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inform. Comput., 256 (2017), pp. 62-82]. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. Together with our previous results providing single-exponential algorithms for particular collections F [J. Baste, I. Sau, and D. M. Thilikos, Theoret. Comput. Sci., 814 (2020), pp. 135-152] and general lower bounds [J. Baste, I. Sau, and D. M. Thilikos, J. Comput. Syst. Sci., 109 (2020), pp. 56-77], our algorithm yields the following complexity dichotomy when F = {H} contains a single connected graph H, assuming the Exponential Time Hypothesis: ƒH(tw) = 2Θ(tw) if H is a contraction of the chair or the banner, and fH(tw) = 2Θ(tw-log tw) otherwise. [ABSTRACT FROM AUTHOR]

A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G. Let mmbs (G) be the size of a maximum (inclusion-wise) minimal blocking set of G. This parameter has recently played an important role in the kernelization of Vertex Cover with structural parameterizations. We provide a panorama of the complexity of computing mmbs parameterized by the natural parameter and the independence number of the input graph. We also consider the closely related parameter mmhs , which is the size of a maximum minimal hitting set of a hypergraph. Finally, we consider the problem of computing mmbs parameterized by treewidth, especially relevant in the context of kernelization. Since a blocking set intersects every maximum-sized independent set of a given graph and properties involving counting the sizes of arbitrarily large sets are typically non-expressible in monadic second-order logic, its tractability does not seem to follow from Courcelle's theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem. [ABSTRACT FROM AUTHOR]

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 kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop-kernels. Informally, this type of kernel 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 question of Boria et al. Discrete Appl Math 196:62–71, 2015. https://doi.org/10.1016/j.dam.2014.06.001), and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on K t -free graphs. Back to the MMVC problem, 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 Erdős–Hajnal property. 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 NP ⊆ coNP / poly . [ABSTRACT FROM AUTHOR]

Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G\S belongs to G. We denote by Ak (G) the set of all graphs that are k-apices of G. In the first paper of this series, we obtained upper bounds on the size of the graphs in the minor-obstruction set of Ak (G), i.e., the minor-minimal set of graphs not belonging to Ak (G). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2poly(k) · n3 and either returns a set S certifying that G ∈ Ak (G), or reports that G ∉ Ak (G). Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of G. In the special case where G excludes some apex graph as a minor, we give an alternative algorithm running in 2poly(k) · n²-time. [ABSTRACT FROM AUTHOR]

We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call these parameters mos (G) and χ odd (G) , respectively. We prove that deciding whether χ odd (G) ≤ q is polynomial-time solvable if q ≤ 2 , and NP-complete otherwise. We provide algorithms in time 2 O (rw) · n O (1) and 2 O (q · rw) · n O (1) to compute mos (G) and to decide whether χ odd (G) ≤ q on n-vertex graphs of rank-width at most rw , respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters. [ABSTRACT FROM AUTHOR]

Given a graph G and a digraph D whose vertices are the edges of G, we investigate the problem of finding a spanning tree of G that satisfies the constraints imposed by D. The restrictions to add an edge in the tree depend on its neighborhood in D. Here, we generalize previously investigated problems by also considering as input functions ℓ and u on E(G) that give a lower and an upper bound, respectively, on the number of constraints that must be satisfied by each edge. The produced feasibility problem is denoted by G-DCST, while the optimization problem is denoted by G-DCMST. We show that G-DCST is NP -complete even if functions ℓ and u are taken under tight assumptions, as well as G and D. On the positive side, we prove two polynomial results, one for G-DCST and another for G-DCMST, and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the ETH. Finally, we prove that other previously studied constrained spanning tree (CST) problems can be modeled within our framework, namely, the Conflict CST, the Forcing CST, the At Least One/All Dependency CST, the Maximum Degree CST, the Minimum Degree CST, and the Fixed-Leaves Minimum Degree CST. [ABSTRACT FROM AUTHOR]

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2 d + 2) -regular graphs and a polynomial algorithm for graphs of maximum degree at most d + 2 . The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [DAM, 2020], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the vertex deletion distance of the input graph to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O ∗ 2 n . [ABSTRACT FROM AUTHOR]

Given a graph G, a properk-coloring of G is a partition c = (S i) i ∈ [ 1 , k ] of V(G) into k stable sets S 1 , ... , S k . Given a weight function w : V (G) → R + , the weight of a color S i is defined as w (i) = max v ∈ S i w (v) and the weight of a coloringc as w (c) = ∑ i = 1 k w (i) . Guan and Zhu (Inf Process Lett 61(2):77–81, 1997) defined the weighted chromatic number of a pair (G, w), denoted by σ (G , w) , as the minimum weight of a proper coloring of G. The problem of determining σ (G , w) has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time n o (log n) unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G, w) and an integer k, is σ (G , w) ≤ ∑ v ∈ V (G) w (v) - k ? This parameterization has been recently considered by Escoffier (in: Proceedings of the 42nd international workshop on graph-theoretic concepts in computer science (WG). LNCS, vol 9941, pp 50–61, 2016), who provided an FPT algorithm running in time 2 O (k log k) · n O (1) , and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time 9 k · n O (1) , and prove that no algorithm in time 2 o (k) · n O (1) exists under the ETH. On the other hand, we present a kernel with at most (2 k - 1 + 1) (k - 1) vertices, and rule out the existence of polynomial kernels unless NP ⊆ coNP / poly , even on split graphs with only two different weights. Finally, we identify classes of graphs allowing for polynomial kernels, namely interval graphs, comparability graphs, and subclasses of circular-arc and split graphs, and in the latter case we present lower bounds on the degrees of the polynomials. [ABSTRACT FROM AUTHOR]

For a finite collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, deciding whether there exists S ⊆ V (G) with |S| ≤ k such that G\S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-Deletion when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F, the smallest function fF such that F-M-DELETION can be solved in time fF(tw) . nO(1) on n-vertex graphs. We prove that fF(tw) = 2²O(tw.logtw) for every collection F, that fF(tw) = 2O(tw.log tw) if F contains a planar graph, and that fF(tw) = 2O(tw) if in addition the input graph G is planar or embedded in a surface. We also consider the version of the problem where the graphs in F are forbidden as topological minors, called F-TM-Deletion. We prove similar results for this problem, except that in the last two algorithms, instead of requiring F to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic. [ABSTRACT FROM AUTHOR]

For two positive integers k and ℓ , a (k × ℓ) -spindle is the union of k pairwise internally vertex-disjoint directed paths with ℓ arcs each between two vertices u and v. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed ℓ ≥ 1 , finding the largest k such that an input digraph G contains a subdivision of a (k × ℓ) -spindle is polynomial-time solvable if ℓ ≤ 3 , and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results. [ABSTRACT FROM AUTHOR]

We study the minimum diameter spanning tree problem under the reload cost model (Diameter‐Tree for short) introduced by Wirth and Steffan. In this problem, given an undirected edge‐colored graph G, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of G of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the Diameter‐Tree problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree Δ of the input graph. We prove that Diameter‐Tree is para‐NP‐hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove Diameter‐Tree to be NP‐hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan proved that the problem can be solved in polynomial time on graphs with Δ = 3, and Galbiati proved that it is NP‐hard if Δ = 4. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP‐hard if Δ = 3, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that Diameter‐Tree is in XP and W[1]‐hard parameterized by the treewidth plus Δ. [ABSTRACT FROM AUTHOR]

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where c ≥ 1 is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer c ≥ 1 , and that Dominating Set does not for any integer c ≥ 2 even on degenerate graphs, unless NP ⊆ coNP / poly . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for c ≥ 3 and an or-cross-composition for c = 2 . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for c = 1 , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter. [ABSTRACT FROM AUTHOR]

Golumbic, Hirst, and Lewenstein define a matching in a simple, finite, and undirected graph G to be uniquely restricted if no other matching covers exactly the same set of vertices. We consider uniquely restricted edge‐colorings of G, defined as partitions of its edge set into uniquely restricted matchings, and study the uniquely restricted chromatic index χ′ur(G) of G, defined as the minimum number of uniquely restricted matchings required for such a partition. For every graph G, χ′(G)≤a′(G)≤χ′ur(G)≤χ′s(G), where χ′(G) is the classical chromatic index, a′(G) the acyclic chromatic index, and χ′s(G) the strong chromatic index of G. While Vizing's famous theorem states that χ′(G) is either the maximum degree Δ(G) of G or Δ(G)+1, two famous open conjectures due to Alon, Sudakov, and Zaks, and to Erdős and Nešetřil concern upper bounds on a′(G) and χ′s(G) in terms of Δ(G). Since χ′ur(G) is sandwiched between these two parameters, studying upper bounds in terms of Δ(G) is a natural problem. We show that χ′ur(G)≤Δ(G)2 with equality if and only if some component of G is KΔ(G),Δ(G). If G is connected, bipartite, and distinct from KΔ(G),Δ(G), and Δ(G) is at least 4, then, adapting Lovász's elegant proof of Brooks' theorem, we show that χ′ur(G)≤Δ(G)2−Δ(G). Our proofs are constructive and yield efficient algorithms to determine the corresponding edge‐colorings. [ABSTRACT FROM AUTHOR]

During the last years, several algorithmic meta-theorems have appeared (Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive from them constructive kernels with reasonably low explicit constants. To fill this gap, we recently presented [STACS 2014] a framework to obtain explicit linear kernels for some families of problems whose solutions can be certified by a subset of vertices. In this article we enhance our framework to deal with packing problems, that is, problems whose solutions can be certified by collections of subgraphs of the input graph satisfying certain properties. F -Packing is a typical example: for a family F of connected graphs that we assume to contain at least one planar graph, the task is to decide whether a graph G contains k vertex-disjoint subgraphs such that each of them contains a graph in F as a minor. We provide explicit linear kernels on sparse graphs for the following two orthogonal generalizations of F -Packing: for an integer ℓ ⩾ 1 , one aims at finding either minor-models that are pairwise at distance at least ℓ in G (ℓ - F -Packing), or such that each vertex in G belongs to at most ℓ minors-models (F -Packing with ℓ -Membership). Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors. [ABSTRACT FROM AUTHOR]

Let Wt denote the wheel on t + 1 vertices. We prove that for every integer t \geq 3 there is a constant c = c(t) such that for every integer k ≥ 1 and every graph G, either G has k vertex-disjoint subgraphs each containing Wt as a minor, or there is a subset X of at most ck log k vertices such that G-X has no Wt minor. This is best possible, up to the value of c. We conjecture that the result remains true more generally if we replace Wt with any fixed planar graph H. [ABSTRACT FROM AUTHOR]

Given two graphs G and H, we define v-coverH(G) (resp. e-coverH(G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also v-packH(G) (resp. e-packH(G)) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H. We denote by θr the graph with two vertices and r parallel edges between them. When H=θr, the parameters v-coverH, e-coverH, v-packH, and e-packH are NP-hard to compute (for sufficiently big values of r). Drawing upon combinatorial results in Chatzidimitriou et al. (Minors in graphs of large θr-girth, 2015, arXiv:1510.03041), we give an algorithmic proof that if v-packθr(G)≤k, then v-coverθr(G)=O(klogk), and similarly for e-packθr and  e-coverθr. In other words, the class of graphs containing θr as a minor has the vertex/edge Erdős-Pósa property, for every positive integer r. Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(logOPT)-approximation algorithm for v-packθr, v-coverθr, e-packθr, and e-coverθr that runs in O(n·log(n)·m) steps. Also, we derive several new Erdős-Pósa-type results from the techniques that we introduce. [ABSTRACT FROM AUTHOR]

The tree-cut width of a graph is a graph parameter defined by Wollan (J Combin Theory, Ser B, 110:47-66, 2015) with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input n-vertex graph G and an integer w, our algorithm either confirms that the tree-cut width of G is more than w or returns a tree-cut decomposition of G certifying that its tree-cut width is at most 2 w, in time $$2^{O(w^2\log w)} \cdot n^2$$ . Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known. [ABSTRACT FROM AUTHOR]

For two integers r, ℓ ≥ 0, a graph G = ( V, E) is an ( r, ℓ)-graph if V can be partitioned into r independent sets and ℓ cliques. In the parameterized ( r, ℓ)- Vertex Deletion problem, given a graph G and an integer k, one has to decide whether at most k vertices can be removed from G to obtain an ( r, ℓ)-graph. This problem is NP-hard if r + ℓ ≥ 1 and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of ( r, ℓ)- Vertex Deletion was known for all values of ( r, ℓ) except for (2,1), (1,2), and (2,2). We prove that each of these three cases is FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of k. We consider as well the version of ( r, ℓ)- Vertex Deletion where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem. [ABSTRACT FROM AUTHOR]

In phylogenetics, a central problem is to infer the evolutionary relationships between a set of species X; these relationships are often depicted via a phylogenetic tree-a tree having its leaves labeled bijectively by elements of X and without degree-2 nodes-called the 'species tree.' One common approach for reconstructing a species tree consists in first constructing several phylogenetic trees from primary data (e.g., DNA sequences originating from some species in X), and then constructing a single phylogenetic tree maximizing the 'concordance' with the input trees. The obtained tree is our estimation of the species tree and, when the input trees are defined on overlapping-but not identical-sets of labels, is called 'supertree.' In this paper, we focus on two problems that are central when combining phylogenetic trees into a supertree: the compatibility and the strict compatibility problems for unrooted phylogenetic trees. These problems are strongly related, respectively, to the notions of 'containing as a minor' and 'containing as a topological minor' in the graph community. Both problems are known to be fixed parameter tractable in the number of input trees k, by using their expressibility in monadic second-order logic and a reduction to graphs of bounded treewidth. Motivated by the fact that the dependency on k of these algorithms is prohibitively large, we give the first explicit dynamic programming algorithms for solving these problems, both running in time $$2^{O(k^2)} \cdot n$$ , where n is the total size of the input. [ABSTRACT FROM AUTHOR]

A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs. In particular, we achieve a ratio of 5/9 for subcubic bipartite graphs, improving over a 1/2-approximation algorithm proposed by Mishra. Furthermore, we study the uniquely restricted chromatic index of a graph, defined as the minimum number of uniquely restricted matchings into which its edge set can be partitioned. We provide tight upper bounds in terms of the maximum degree and characterize all extremal graphs. Our constructive proofs yield efficient algorithms to determine the corresponding edge colorings. [ABSTRACT FROM AUTHOR]

Let be the number of labeled graphs on n vertices and treewidth at most k (equivalently, the number of labeled partial k-trees). We show that ]> ]> k > 1 ]> ]> ]> c > 0 ]> ]> ]> ( log k ) n ]> . The upper bound is a direct consequence of the well-known formula for the number of labeled k-trees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most k. [ABSTRACT FROM AUTHOR]

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [(Meta) kernelization, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2009, pp. 629– 638] on graphs of bounded genus, then generalized by Fomin et al. [Bidimensionality and kernels, in Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadephia, 2010, pp. 503–510] to graphs excluding a fixed minor, and by Kim et al. [Linear kernels and single-exponential algorithms via protrusion decompositions, in Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 7965 (2013), pp. 613–624] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions, but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper, we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in counting monadic second order logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for r-Dominating Set and r-Scattered Set on apex-minor-free graphs, and for PLANAR-F-DELETION on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected. [ABSTRACT FROM AUTHOR]