International audienceWe present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X⊆V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has a finite integer index and such that Yes-instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar-F-Deletion asks whether G has a set X⊆V(G) such that |X| ⩽ k and G − X is H-minor-free for every H ε F. As our second application, we present the first single-exponential algorithm to solve Planar-F-Deletion. Namely, our algorithm runs in time 2O(k) · n2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F

International audienceWe present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs.Let Fbe a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar F- Deletion asks whether G has a set X ⊆ V(G) such that |X|⩽k and G − X is H-minor-free for every H∈F. As our second application, we present the first single-exponential algorithm to solve Planar F- Deletion. Namely, our algorithm runs in time 2 O(k)·n 2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F

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Abstract. We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf Out-Tree and Directed Maximum Leaf Spanning Out-Tree in the case of directed graphs. The run time of our algorithm is O(poly(|V |) + 4 k k 2 ) on undirected graphs, and O(4 k |V |·|E|) on directed graphs. Currently, the fastest algorithms for these problems have run times of O(poly(n) + 6.75 k poly(k)) and 2 O(k log k) poly(n), respectively.

In this survey, we review practical algorithms for graph-theoretic problems that are expressible in monadic second-order logic. Monadic second-order (MSO) logic allows quantifications over unary relations (sets) and can be used to express a host of useful graph properties such as connectivity, c-colorability (for a fixed c), Hamiltonicity and minor inclusion. A celebrated theorem in this area by Courcelle states that any graph problem expressible in MSO can be solved in linear time on graphs that admit a tree-decomposition of constant width. Courcelle's Theorem has been used thus far as a theoretic tool to establish that linear-time algorithms exist for graph problems by demonstrating that the problem in question is expressible by an MSO formula. A straightforward implementation of the algorithm in the proof of Courcelle's Theorem is useless as it runs into space-explosion problems even for small values of treewidth. Of late, there have been several attempts to circumvent these problems and we review some of these in this survey. This survey also introduces the reader to the notions of tree-decompositions and the basics of monadic second order logic.

Abstract. In contrast to undirected width measures (such as treewidth or clique-width), which have provided many important algorithmic applications, analogous measures for digraphs such as DAGwidth or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer-Ordyniak, Dankelmann-Gutin-Kim, or LampisKaouri-Mitsou, have given some evidence for this. We support this direction by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAGdepth. On the positive side, we also note that taking Kanté's directed generalization of rank-width as a parameter makes many problems fixed parameter tractable.

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic secondorder logic with edge-set quantifications (i.e., MSO 2 model-checking) is decidable in linear time on any class of graphs of bounded tree-width. Recently, Kreutzer and Tazari [15] proved a corresponding complexity lower-bound-that MSO 2 model-checking is not even in XP wrt. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is polylogarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).In this paper we present a closely related result. We show that even MSO 1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari; (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO 1 -definable properties on vertex labeled graphs instead of MSO 2 -definable properties on unlabeled graphs.Our result has an interesting consequence in the realm of digraph width measures: Strengthening the recent result [10], we show that no subdigraphmonotone measure can be "algorithmically useful", unless it is within a polylogarithmic factor of undirected tree-width.

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