Geevarghese Philip

On the Kernelization Complexity of Colorful Motifs

The Colorful Motif problem asks if, given a vertex-colored graph G, there exists a subset S of vertices of G such that the graph induced by G on S is connected and contains every color in the graph exactly once. The problem is motivated by applications in computational biology and is also well-studied from the theoretical point of view. In particular, it is known to be NP-complete even on trees of maximum degree three [Fellows et al, ICALP 2007]. In their pioneering paper that introduced the color-coding technique, Alon et al. [STOC 1995] show, inter alia, that the problem is FPT on general graphs. More recently, Cygan et al. [WG 2010] showed that Colorful Motif is NP-complete on comb graphs, a special subclass of the set of trees of maximum degree three. They also showed that the problem is not likely to admit polynomial kernels on forests.

Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels

We show that the k -Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K i,j as a subgraph, for any fixed i,j ≥ 1. This strictly includes every class of graphs for which this problem has been previously shown to have FPT algorithms and/or polynomial kernels. In particular, our result implies that the problem restricted to bounded-degenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner in [3].

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F⊆V, |F|≤k, such that G[V∖F] is a forest and G[F] is connected.We show that Connected Feedback Vertex Set can be solved in time O(2O(k)nO(1)) on general graphs and in time

Hitting Forbidden Minors: Approximation and Kernelization

O(2^{O(\sqrt{k}\log k)}n^{O(1)})

Point Line Cover: The Easy Kernel is Essentially Tight

on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.

A quartic kernel for pathwidth-one vertex deletion

We study a general class of problems called F-Deletion problems. In an F-Deletion problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-Deletion problem when F contains a planar graph. We give - a linear vertex kernel on graphs excluding t-claw K_(1,t), the star with t leaves, as an induced subgraph, where t is a fixed integer. - an approximation algorithm achieving an approximation ratio of O(log^(3/2) OPT), where

FPT algorithms for connected feedback vertex set

OPT

Beyond Max-Cut: Lambda-extendible Properties Parameterized above the Poljak-Turzík Bound

is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F only contains graph theta_c as a minor for a fixed integer c. The graph theta_c consists of two vertices connected by

Algorithmic aspects of dominator colorings in graphs

c

A single-exponential FPT algorithm for the K 4 -minor cover problem

parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.