A Polynomial Kernel for Block Graph Deletion

Kim, Eun Jung; Kwon, O-joung

doi:10.1007/s00453-017-0316-2

Cited by 41 publications

(16 citation statements)

References 33 publications

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“…Kim and Kwon [11] gave an FPT algorithm with running time O ⋆ (10 k ) and a kernel of size O(k 9 ) for the BGVD problem. In this paper we improve both these results via a novel connection to Feedback Vertex Set problem.…”

Section: Block Graph Vertex Deletion (Bgvd)mentioning

confidence: 99%

“…In the algorithm that we give for the BGVD problem, as a sub-routine we use the algorithm for the Weighted-FVS problem. For obtaining a better polynomial kernel for BGVD, most of our Reduction Rules are same as those used in [11]. On the way to our result we also design a factor four approximation algorithm for BGVD.…”

Section: Block Graph Vertex Deletion (Bgvd)mentioning

confidence: 99%

“…The Reduction rules BGVD.1 to BGVD.5 are safe and can be applied in polynomial time [11]. For sake of clarity we denote the reduced instance at each step by (G, k).…”

Section: Improved Kernel For Block Graph Vertex Deletionmentioning

confidence: 99%

“…However, there are several other notions similar to treewidth where several NP-hard problems become polynomial time solvable; these include cliquewidth, rankwidth and linear rankwidth. Recently, Kanté et al [10] studied the problem of deleting k vertices to get a graph of linear rankwidth one and in an other study Kim and Kwon [11] studied deleting k vertices to get a block graph.…”

Section: Introductionmentioning

confidence: 99%

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A Faster FPT Algorithm and a Smaller Kernel for Block Graph Vertex Deletion

Agrawal

Kolay²,

Lokshtanov

et al. 2016

LATIN 2016: Theoretical Informatics

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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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Section: Block Graph Vertex Deletion (Bgvd)mentioning

confidence: 99%

Section: Block Graph Vertex Deletion (Bgvd)mentioning

confidence: 99%

“…The Reduction rules BGVD.1 to BGVD.5 are safe and can be applied in polynomial time [11]. For sake of clarity we denote the reduced instance at each step by (G, k).…”

Section: Improved Kernel For Block Graph Vertex Deletionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Faster FPT Algorithm and a Smaller Kernel for Block Graph Vertex Deletion

Agrawal

Kolay²,

Lokshtanov

et al. 2016

LATIN 2016: Theoretical Informatics

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“…Now, we discuss a "sunflower structure" that allows us to find a vertex that can be safely removed. A similar technique was used in [1,13,21]; there, Gallai's A-path Theorem is used to find many obstructions whose pairwise intersections are exactly one vertex; here, we use different objects to achieve the same thing.…”

Section: Polynomial Kernelsmentioning

confidence: 99%

Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

Bonnet

Brettell

Kwon

et al. 2016

Graph-Theoretic Concepts in Computer Science

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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 2 O(k log d) n O(1) . When P contains all split graphs, we show that this running time is essentially optimal unless the Exponential Time Hypothesis fails. On the other hand, if P consists of only complete graphs, or only cycle graphs and K2, then Bounded P-Block Vertex Deletion admits a c k n O(1) -time algorithm for some constant c independent of d. We also show that Bounded P-Block Vertex Deletion admits a kernel with O(k 2 d 7 ) vertices.1 A block graph is the usual name in the literature for a graph where each block is a complete subgraph. However, since we are dealing here with both blocks and block graphs, to avoid confusion we instead use the term complete-block graph and call the corresponding vertex deletion problem Complete Block Vertex Deletion.

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