Elimination Distance to Bounded Degree on Planar Graphs

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)? n^c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k

First-Order Logic with Connectivity Operators

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates $conn_k (x, y, z_1 ,\ldots, z_k)$ that hold true in a graph if there exists a path between (the valuations of) $x$ and $y$ after (the valuations of) $z_1,\ldots,z_k$ have been deleted, we obtain separator logic $FO + conn$. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic $FO + DP$ by adding the atomic predicates $disjoint-paths_k [(x_1, y_1 ),\ldots , (x_k , y_k )]$ that evaluate to true if there are internally vertex disjoint paths between (the valuations of) $x_i$ and $y_i$ for all $1 \le i \le k$. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Finally, we compare the expressive power of the new logics with that of transitive closure logics and monadic second-order logic.Comment: 18 pages, 3 figure

Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets. A blocking set is a set of vertices such that no optimal vertex cover contains all vertices in the blocking set, and the study of minimal blocking sets played implicit and explicit roles in many existing results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class ?, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class ?, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes ?, including the class ?_{LP} of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or ?_{LP} graphs

Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ?, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ? for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ?, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ? for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ? has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number