Abstract. A graph G is called a block graph if each maximal 2-connected component of G is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph G and a positive integer k and the objective to check whether there exists a subset S ⊆ V (G) of size at most k such that the graph induced on V (G) \ S is a block graph. In this paper we give an FPT algorithm with running time 4 k n O(1) and a polynomial kernel of size O(k 4 ) for Block Graph Vertex Deletion. The running time of our FPT algorithm improves over the previous best algorithm for the problem that ran in time 10 k n O(1) and the size of our kernel reduces over the previously known kernel of size O(k 9 ). Our results are based on a novel connection between Block Graph Vertex Deletion and the classical Feedback Vertex Set problem in graphs without induced C4 and K4 − e. To achieve our results we also obtain an algorithm for Weighted Feedback Vertex Set running in time 3.618 k n O(1) and improving over the running time of previously known algorithm with running time 5 k n O(1) .

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Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w : V (G) → R + , find a minimum weight subset S ⊆ V (G) such that G−S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log O(1) n)approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log O(1) n)-approximation algorithms for the following vertex deletion problems. Let F be a finite set of graphs containing a planar graph, and F = G(F) be the family of graphs such that every graph H ∈ G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F = G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log 1.5 n) and O(log 2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. We give an O(log 2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. We give an O(log 3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(log O(1) n)-approximation algorithms for many other problems as well.

An input to a conflict-free variant of a classical problem Γ, called Conflict-Free Γ, consists of an instance I of Γ coupled with a graph H, called the conflict graph. A solution to Conflict-Free Γ in (I, H) is a solution to I in Γ, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid.

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