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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

TL;DR

The paper studies the F-Vertex Splitting problems under Inclusive and Exclusive variants for four graph classes: constellations, cycle graphs, linear forests, and bipartite graphs. It provides polynomial-time algorithms for cycle graphs and linear forests, while proving NP-hardness for constellations and bipartite graphs, via a Gamma-decomposition framework and reductions to Weighted Star Decomposition and Bipartite Vertex Deletion. A key contribution is establishing equivalences between Inclusive/Exclusive Vertex Splitting and deletion problems, enabling transfer of kernelization and FPT results. The results offer a near-complete dichotomy for these simple graph classes and open avenues for further exploration on other classes and parameterized complexity.

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.

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

TL;DR

The paper studies the F-Vertex Splitting problems under Inclusive and Exclusive variants for four graph classes: constellations, cycle graphs, linear forests, and bipartite graphs. It provides polynomial-time algorithms for cycle graphs and linear forests, while proving NP-hardness for constellations and bipartite graphs, via a Gamma-decomposition framework and reductions to Weighted Star Decomposition and Bipartite Vertex Deletion. A key contribution is establishing equivalences between Inclusive/Exclusive Vertex Splitting and deletion problems, enabling transfer of kernelization and FPT results. The results offer a near-complete dichotomy for these simple graph classes and open avenues for further exploration on other classes and parameterized complexity.

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.

Paper Structure

This paper contains 10 sections, 22 theorems, 22 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Vertex Splitting Operation and Problem Definitions
  3. Our Results
  4. Preliminaries
  5. Gamma-Decomposition Framework and its connection to Vertex Splitting
  6. Constellation-Inclusive/Exclusive Vertex Splitting
  7. Cycle Graph-Inclusive/Exclusive Vertex Splitting
  8. Linear Forest-Inclusive/Exclusive Vertex Splitting
  9. Bipartite Inclusive/Exclusive Vertex Splitting
  10. Conclusion

Key Result

Lemma 1.1

Let $G=(V, E)$ be a graph without isolated vertices, and let $\mathcal{P}$ be a $\Gamma$-decomposition of $G$ such that ${\normalfont\texttt{wgt}}_G(\mathcal{P}) \leq |V(G)|+\alpha$. Then, either $G$ is already a disjoint union of graphs of $\Gamma$ or there exists a vertex $u \in V$ such that $u$ c

Figures (1)

  • Figure 1: On the left: a graph $G$ with a $\Gamma$-decomposition where $\Gamma= \{K_3\}$. The subgraph $H_1$ is shown in blue. On the right: the graph $G'$, obtained by splitting $u$ into $u_{\text{in}}$ and $u_{\text{out}}$, along with a corresponding $\Gamma$-decomposition of $G'$.

Theorems & Definitions (34)

  • Definition 1.1: Inclusive Vertex Splitting
  • Definition 1.2: Exclusive Vertex Splitting
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.1
  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 24 more