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Parameterized Algorithms for Editing to Uniform Cluster Graph

Authors: Gaikwad, Ajinkya; Kumar, Hitendra; Maity, Soumen;

Parameterized Algorithms for Editing to Uniform Cluster Graph

Abstract

We study the parameterized complexity of transforming graphs into Uniform Cluster graphs, where each component is an equal-sized clique. We consider Uniform Cluster Vertex Deletion (UCVD), Uniform Cluster Edge Deletion (UCED), Uniform Cluster Edge Addition (UCEA), Uniform Cluster Edge Editing (UCEE), Uniform Cluster Exclusive Vertex Splitting (UCEVS), and Uniform Cluster Inclusive Vertex Splitting (UCIVS). For UCVD, we provide a vertex kernel of size $\mathcal{O}(k^{3})$ and an FPT algorithm with running time $2^{k} \cdot n^{\mathcal{O}(1)}$, improving the known $3^{k} \cdot n^{\mathcal{O}(1)}$ algorithm. For edge-based variants, we obtain a $\mathcal{O}(k^{2})$ vertex kernel for UCEE and linear vertex kernels for UCED and UCEA, improving the best-known results. Additionally, we present a $1.47^{k} \cdot n^{\mathcal{O}(1)}$ algorithm for UCED, improving upon the previous $2^{k} \cdot n^{\mathcal{O}(1)}$ bound. We develop a sub-exponential algorithm for UCED on everywhere dense graphs by reducing it to $d$-Way Cut. Lastly, we study vertex splitting operations and provide vertex kernels of size $4k$ for both UCIVS and UCEVS.

Keywords

FOS: Computer and information sciences, Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms, Data Structures and Algorithms (cs.DS), Computational Complexity (cs.CC)

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
Green