On the Complexity of Vertex-Splitting Into an Interval Graph

TL;DR

The paper investigates vertex splitting as a graph modification to obtain interval graphs, proving that deciding whether a graph can be transformed into an interval graph with at most $k$ splits is NP-hard, even for subcubic planar bipartite inputs. It provides a sharp positive result: transforming any graph into a disjoint union of paths can be done optimally in polynomial time, and triangle-free graphs can be split into unit interval graphs with polynomial-time algorithms. The work also analyzes how splitting compares to edge and vertex deletions toward chordal graphs, showing fundamental separations between these modification types and uncovering nuanced behavior under independence constraints. Together, these results establish both hardness and tractability borders for splitting-based graph modification and open several directions for future study, including chordal and unit interval vertex splitting and related parameterized questions.

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.

Figures (8)

• Figure 1: The claw, net and tent graphs.
• Figure 2: The graph $T_2$.
• Figure 3: A planar cubic graph $G$ and the resulting graph $G'$ of the reduction.
• Figure 4: The resulting graph after the flipping sequence.
• Figure 5: The graph $G_k$ for $k=3$. This graph satisfies $ChVS(G_k) = 1$ and $ChED(G_k) \geq k$ and $\alpha(G_k) = 2$.

Theorems & Definitions (30)

• lemma 1
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• lemma 2
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• corollary 1
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• theorem 1
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• corollary 2
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