A polynomial kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph $G$ on $n$ vertices and a positive integer $k$, and the objective is to check whether it is possible to delete at most $k$ vertices from $G$ to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $\mathcal{O}(k^{6})$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of `complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $10^{k}\cdot n^{\mathcal{O}(1)}$.Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201

Parameterized vertex deletion problems for hereditary graph classes with a block property

For a class of graphs P, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices such that each block of G − S has at most d vertices and is in P. We show that when P satisfies a natural hereditary property and is recognizable in polynomial time, Bounded P-Block Vertex Deletion can be solved in time 2O(k log d)nO(1), and this running time cannot be improved to 2o(k log d)nO(1), in general, unless the Exponential Time Hypothesis fails. On the other hand, if P consists of only complete graphs, or only K1,K2, and cycle graphs, then Bounded P-Block Vertex Deletion admits a cknO(1)-time algorithm for some constant c independent of d. We also show that Bounded P-Block Vertex Deletion admits a kernel with O(k2d7) vertices. © Springer-Verlag GmbH Germany 2016

An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of linear rankwidth at most $1$ and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $f(k)\cdot n^3$ for some function $f$, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $2^{\mathcal{O}(k)}\cdot n^4$. We also prove that the running time cannot be improved to $2^{o(k)}\cdot n^{\mathcal{O}(1)}$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201

Exploring Subexponential Parameterized Complexity of Completion Problems

Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem, we are given a graph $G$ and an integer $k$ as input, and asked whether at most $k$ edges can be added to $G$ so that the resulting graph does not contain a graph from ${\cal F}$ as an induced subgraph. It appeared recently that special cases of ${\cal F}$-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding to the case of ${\cal F}=\{C_4,C_5,C_6,\ldots\}$, and the problem of completing into a split graph, i.e., the case of ${\cal F}=\{C_4, 2K_2, C_5\}$, are solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this phenomenon is the main motivation for our research on ${\cal F}$-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal F}$-Completion for ${\cal F} =\{C_4, P_4\}$, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where ${\cal F} = \{2K_2, C_4\}$, and Threshold Completion, where ${\cal F} = \{2K_2, P_4, C_4\}$, are also solvable in time $2^{O(\sqrt{k}\log{k})} n^{O(1)}$. We complement our algorithms for ${\cal F}$-Completion with the following lower bounds: - For ${\cal F} = \{2K_2\}$, ${\cal F} = \{C_4\}$, ${\cal F} = \{P_4\}$, and ${\cal F} = \{2K_2, P_4\}$, ${\cal F}$-Completion cannot be solved in time $2^{o(k)} n^{O(1)}$ unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of ${\cal F}$-Completion problems for ${\cal F}\subseteq\{2K_2, C_4, P_4\}$.Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in the proceedings of STACS'1

Fast Dynamic Graph Algorithms for Parameterized Problems

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are $o(|G|)$, that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT). We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size $k$. These dynamic graphs achieve almost the best possible update time $O(\mathrm{poly}(k)\log n)$ and the query time $O(f(\mathrm{poly}(k),k))$, where $f(n,k)$ is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem. We also give a dynamic graph for Chromatic Number parameterized by the solution size of Cluster Vertex Deletion, and a dynamic graph for bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming the parameter is a constant, each dynamic graph can be updated in $O(\log n)$ time and can compute a solution in $O(1)$ time. These results are obtained by another approach.Comment: SWAT 2014 to appea