Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai

Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

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

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution

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