For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T be the family of graphs such that each graph in T can be made into a tree by deleting at most edges. Thus, the problem we study is T -Contraction. We design an FPT algorithm for T -Contraction running in time O((2 √ + 2) O(k+ ) · n O(1) ). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T -Contraction of size O([k(k + 2 )] ( α α−1 +1) ).

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In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Π-Deletion problem which is the dynamic variant of the Π-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Π-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740 k n O(1) (polynomial space) and 1.1277 k n O(1) (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667 k n O(1) running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2 k n O(1) running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

In the precoloring extension problem (PREXT) a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper coloring of the graph with the given number of colors. Two parameterized versions of the problem are studied in the paper: either the number of precolored vertices or the number of colors used in the precoloring is restricted to be at most k. We show that for chordal graphs these problems are polynomial-time solvable for every fixed k, but W[1]-hard if k is the parameter. For a graph class F, let F + ke (resp., F + kv) denote those graphs that can be made to be a member of F by deleting at most k edges (resp., vertices). We investigate the connection between PREXT in F (with the two parameters defined above) and the coloring of F + ke, F + kv graphs (with k being the parameter). Answering an open question of Leizhen Cai [Parameterized complexity of vertex colouring, Discrete Appl. Math. 127 (2003) 415-429], we show that coloring chordal+ke graphs is fixed-parameter tractable.