Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.26938

Inclusive and Exclusive Vertex Splitting into Specific Graph Classes: NP Hardness and Algorithms

Ajinkya Gaikwad·Hitendra Kumar·S. Padmapriya·Praneet Kumar Patra·Harsh Sanklecha·Soumen Maity
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a family of graph modification problems called the F-Vertex Splitting problem. Given a graph G, the task is to determine whether G can be transformed into a graph G-prime belonging to a graph class F through a sequence of at most k vertex splits. We investigate this problem for several target graph classes, namely constellations, cycle graphs, linear forests, and bipartite graphs. We analyze both inclusive and exclusive variants of vertex splitting, as introduced by Abu-Khzam and collaborators (ISCO 2018). Our results show that the F-Vertex Splitting problem is polynomial-time solvable when F is a cycle graph or a linear forest, for both variants. In contrast, when F is a constellation or a bipartite graph, the problem is NP-complete for both variants.

Explore related subjects

cs.DScs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ajinkya Gaikwad, Hitendra Kumar, S. Padmapriya, Praneet Kumar Patra, Harsh Sanklecha, Soumen Maity. 2025-10-30. Inclusive and Exclusive Vertex Splitting into Specific Graph Classes: NP Hardness and Algorithms. https://arxiv.org/abs/2510.26938

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗