Abstract In the DOMINATING SET problem we are given an n-vertex graph G with a positive integer k and we ask whether there exists a vertex subset D of size at most k such that every vertex of G is either in D or is adjacent to some vertex of D. In the connected variant, CONNECTED DOMINATING SET, we also demand the subgraph induced by D to be connected. Both variants are basic graph problems, known to be NP-complete, and many algorithmic approaches have been tried on them. In this paper we study both problems on graphs excluding a fixed graph H as a minor from the kernelization point of view. Our main results are polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G 0 on O(k) vertices such that G has a (connected) dominating set of size k if and only if G 0 has. The polynomial time algorithm that obtains such equivalent instance is known as kernelization algorithm and its output is called a problem kernel. If the size of the output can be bounded by a polynomial (linear) function of k, then it is called polynomial (linear) kernel. Prior to our work, the only polynomial kernel for DOMINATING SET on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009 and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is k c(H) , where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for DOMINATING SET on H-minor free graphs. We answer this question in affirmative. For CONNECTED DOMINATING SET no polynomial kernel on H-minor free graphs was known prior to our work. As a byproduct of our results we also obtain the first subexponentail time algorithm for CON-NECTED DOMINATING SET running in time 2

Abstract. The theory of Graph Minors by Robertson and Seymour is one of the deepest and significant theories in modern Combinatorics. This theory has also a strong impact on the recent development of Algorithms, and several areas, like Parameterized Complexity, have roots in Graph Minors. Until very recently it was a common believe that Graph Minors Theory is of mainly theoretical importance. However, it appears that many deep results from Robertson-Seymour’s theory can be also used in the design of practical algorithms. Minor containment testing is one of the most algorithmically important and technical parts of the theory, and minor containment in graphs of bounded branchwidth is the basic ingredient of this algorithm. In order to implement minor containment testing on graphs of bounded branchwidth, Hicks [NETWORKS 04] described an algorithm, that in time O(3 k2 · (h + k − 1)! · m) decides if a graph G with m edges and branchwidth k, contains a fixed graph H on h vertices as a minor. That algorithm follows the ideas introduced by Robertson and Seymour in [J’CTSB 95]. In this work we improve the dependence on k of Hicks ’ result by showing that checking if H is a minor of G can be done in time O(2 (2k+1)·log k · h 2k · 2 2h2 · m). Our approach is based on a combinatorial object called

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In the Connected Dominating Set problem we are given as input a graph G and a positive integer k, and are asked if there is a set S of at most k vertices of G such that S is a dominating set of G and the subgraph induced by S is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer k (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function g(k). The new instance is called a g(k) kernel for the problem. If g(k) is a polynomial in k then we say that the problem admits polynomial kernels. The girth of a graph G is the length of a shortest cycle in G. It turns out that Connected Dominating Set is “hard ” on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set does not have a kernel of any size on graphs of girth 3 or 4 (since the problem is W[2]-hard); admits a g(k) kernel, where g(k) is k O(k) , on graphs of girth 5 or 6 but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; has a cubic (O(k 3)) kernel on graphs of girth at least 7. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs. Digital Object Identifier 10.4230/LIPIcs.xxx.yyy.p 1

In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3 k n O(1). Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time cannot be improved to 2 o(k) n O(1). We show that OCT admits a randomized algorithm running in O(n O(1) + 2 O( √ k log k) n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(2 O(k log k) n) time. This improves over an algorithm of Fiorini et

Abstract. The dominating set problem has been extensively studied in the realm of parameterized complexity. It is one of the most common sources of reductions while proving the parameterized intractability of problems. In this paper, we look at dominating set and its generalization r-dominating set on graphs of bounded diameter in the realm of parameterized complexity. We show that Dominating set remains W[2]-hard on graphs of diameter 2, while r-dominating set remains W[2]-hard on graphs of diameter r +1. The lower bound on the diameter in our intractability results is the best possible, as r-dominating set is clearly polynomial time solvable on graphs of diameter at most r. 1

We analyze a common feature of p-Kemeny AGGregation (p-KAGG) and p-One-Sided Crossing Minimization (p-OSCM) to provide new insights and findings of interest to both the graph drawing community and the social choice community. We obtain parameterized subexponential-time algorithms for p-KAGG—a problem in social choice theory—and for p-OSCM—a problem in graph drawing. These algorithms run in timeO.2O. p k logk//, where k is the parameter, and significantly improve the previous best algorithms with running times O.1.403k/ and O.1.4656k/, respectively. We also study natural “above-guarantee ” versions of these problems and show them to be fixed parameter tractable. In fact, we show that the above-guarantee versions of these problems are equivalent to a weighted variant of p-directed feedback arc set. Our results for the above-guarantee version of p-KAGG reveal an interesting contrast. We show that when the number of “votes ” in the input to p-KAGG is odd the above guarantee version can still be solved in time O.2O. p k logk//, while if it is even then the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails (equivalently, unless FPT = M[1]).