On the Complexity of Vertex-Splitting Into an Interval Graph

TL;DR

This paper investigates the computational complexity of transforming arbitrary graphs into interval graphs through vertex splitting, proving NP-hardness in general but providing polynomial algorithms for specific graph classes.

Contribution

It introduces vertex splitting as a new graph modification operation for interval graphs and establishes NP-hardness of the related decision problem.

Findings

Deciding vertex splitting into interval graphs is NP-hard for subcubic planar bipartite graphs.

Polynomial-time algorithms exist for splitting triangle-free graphs into unit interval graphs.

Efficiently transforming graphs into disjoint paths via vertex splits is also polynomial-time solvable.

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.