On the Complexity of Vertex-Splitting Into an Interval Graph

2026-02-04

AI Summary

This study investigates the computational complexity of transforming a general graph into an interval graph using at most $k$ vertex splitting operations. As the first systematic exploration of vertex splitting for constructing interval graphs, it reveals fundamental differences between this operation and traditional vertex or edge deletion approaches. By integrating techniques from computational complexity theory, structural graph analysis, and algorithm design, the paper establishes that the problem is NP-hard even on subcubic planar bipartite graphs. On the positive side, it presents polynomial-time algorithms for two special cases: transforming a graph into a disjoint union of paths, and converting a triangle-free graph into a unit interval graph.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Vertex splitting is a graph modification operation in which a vertex is replaced by multiple vertices such that the union of their neighborhoods equals the neighborhood of the original vertex. We introduce and study vertex splitting as a graph modification operation for transforming graphs into interval graphs. Given a graph $G$ and an integer $k$, we consider the problem of deciding whether $G$ can be transformed into an interval graph using at most $k$ vertex splits. We prove that this problem is NP-hard, even when the input is restricted to subcubic planar bipartite graphs. We further observe that vertex splitting differs fundamentally from vertex and edge deletions as graph modification operations when the objective is to obtain a chordal graph, even for graphs with maximum independent set size at most two. On the positive side, we give a polynomial-time algorithm for transforming, via a minimum number of vertex splits, a given graph into a disjoint union of paths, and that splitting triangle free graphs into unit interval graphs is also solvable in polynomial time.

Problem

• vertex splitting
• interval graph
• graph modification
• NP-hard
• subcubic planar bipartite graphs

Innovation

• vertex splitting
• interval graph
• NP-hardness
• polynomial-time algorithm
• graph modification