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.