Algorithms and Complexity of s-Club Cluster Vertex Deletion

Chakraborty, Dibyayan; Chandran, L. Sunil; Padinhatteeri, Sajith; Pillai, Raji R.

doi:10.1007/978-3-030-79987-8_11

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…We note that our techniques deviate significantly from the ones in the previous literature [9,10,32]. We show that the optimal solution (for s-CVD on interval graphs) must be one of ''four types'' and the optimum for each of the ''four types'' can be found by solving s-CVD on O(m + n) many induced subgraphs.…”

Section: Definition 1 ([1]mentioning

confidence: 85%

s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

Chakraborty

Chandran

Padinhatteeri

et al. 2022

Graph-Theoretic Concepts in Computer Science

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In this paper, we study the computational complexity of s-Club Cluster Vertex Deletion. Given a graph, s-Club Cluster Vertex Deletion (s-CVD) aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most s. When s = 1, the corresponding problem is popularly known as Cluster Vertex Deletion (CVD). We provide a faster algorithm for s-CVD on interval graphs. For each s ≥ 1, we give an O(n(n+m))-time algorithm for s-CVD on interval graphs with n vertices and m edges. In the case of s = 1, our algorithm is a slight improvement over the O(n 3 )-time algorithm of Cao et al. ( 2018), and for s ≥ 2, it significantly improves the state-of-the-art running time. We also give a polynomial-time algorithm to solve CVD on well-partitioned chordal graphs, a graph class introduced by Ahn et al. (WG 2020) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the Weighted Bipartite Vertex Cover. This generalises a result of Cao et al. (2018) on split graphs. We also show that for any even integer s ≥ 2, s-CVD is NP-hard on well-partitioned chordal graphs.

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Section: Definition 1 ([1]mentioning

confidence: 85%

s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

Chakraborty

Chandran

Padinhatteeri

et al. 2022

Graph-Theoretic Concepts in Computer Science

Self Cite

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“…It is noticeable that there are only a few known cases where the problem can be solved efficiently: cluster-vd is polynomially solvable on block graphs, split graphs and interval graphs [3], and on graphs of bounded treewidth [29]. On the other hand, the complexity status of cluster-vd on many well-studied graph classes is still open, e.g., chordal graphs discussed in [3] and planar bipartite graphs mentioned in [4].…”

Section: Cluster-vdmentioning

confidence: 99%

“…Fig. 4 An example of the reduction from cluster-vd to connected cluster-vd: A bipartite graph G (left) and the bipartite graph G(3) (right) obtained from G and H (3,4,3); the bipartition of the vertex set is indicated by circle and rectangle vertices Theorem 15 For any given integer g ≥ 3, connected cluster-vd is NP-complete on bipartite graphs of maximum degree at most 4 and with girth > g and, assuming ETH, cannot be solved in 2 o( √ n) time.…”

Section: Theorem 12mentioning

confidence: 99%

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

Le,

2024

Theory Comput Syst

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The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph $$G-S$$ G - S is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is $$\textsf{NP}$$ NP -complete. Moreover, in the $$\textsf{NP}$$ NP -completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

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“…A similar approach has been considered in [6] to analyze social networks. The s-club model has also been applied to edit a graph into disjoint clusters (s-clubs) [11,18,32]. A 1-club is a clique, so a natural step towards generalizing cliques using distances is to study the s = 2 case, especially given that 2-clubs have applications in social network analysis and bioinformatics [1,4,31,34,35,44].…”

Section: Introductionmentioning

confidence: 99%

On the Tractability of Covering a Graph with 2-Clubs

Dondi

Lafond

2022

Algorithmica

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Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science, for example when the cohesive subgraph model considered is a clique. In this paper, we consider as a model of cohesive subgraph the 2-clubs, which are induced subgraphs of diameter at most 2. We prove new complexity results on the $$\mathsf {Min~2\text {-}Club~Cover}$$ Min 2 - Club Cover problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs. First, we answer an open question on the decision version of $$\mathsf {Min~2\text {-}Club~Cover}$$ Min 2 - Club Cover that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of $$\mathsf {Min~2\text {-}Club~Cover}$$ Min 2 - Club Cover on some graph classes. We prove that $$\mathsf {Min~2\text {-}Club~Cover}$$ Min 2 - Club Cover remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution, and fixed-parameter tractable on graphs having bounded treewidth.

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Algorithms and Complexity of s-Club Cluster Vertex Deletion

Cited by 5 publications

References 25 publications

s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

On the Tractability of Covering a Graph with 2-Clubs

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