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On the Complexity of Establishing Hereditary Graph Properties via Vertex Splitting

Alexander Firbas, Manuel Sorge

TL;DR

This work analyzes the complexity of turning graphs into members of a hereditary property via vertex splits, formalized as $Π$-Vertex Splitting ($Π$-VS). It yields a tight dichotomy for properties defined by forbidden induced subgraphs with at most three vertices, and then develops a general reduction framework (via $ ext{Constr}(igvec{G},C,S)$ and admissible splitting configurations) to prove NP-hardness for broad families of forbidden subgraphs, including biconnected and higher-connected cases. The authors further show hardness for infinite families of cycles (e.g., Bipartite-VS and Perfect-VS) and examine the parameterized complexity with respect to the number of splits $k$, obtaining para-NP-hardness in some cases and XP algorithms when each vertex can be split at most once. Overall, the paper maps the tractability frontier for $Π$-VS, highlighting the role of subgraph connectivity and cycle structure, and it outlines several open problems on the remaining tractable cases (e.g., $P_4$ and $K_{1,3}$).

Abstract

Vertex splitting is a graph operation that replaces a vertex $v$ with two nonadjacent new vertices and makes each neighbor of $v$ adjacent with one or both of the introduced vertices. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we explore the computational complexity of achieving a given graph property $Π$ by a limited number of vertex splits, formalized as the problem $Π$ Vertex Splitting ($Π$-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of $Π$-VS for graph properties characterized by forbidden subgraphs of size at most 3. Second, we provide a framework that allows to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Leveraging this framework we show NP-completeness when $Π$ is characterized by forbidden subgraphs that are sufficiently well connected. In particular, we show that $F$-Free-VS is NP-complete for each biconnected graph $F$. Third, we study infinite families of forbidden subgraphs, obtaining NP-hardness for Bipartite-VS and Perfect-VS. Finally, we touch upon the parameterized complexity of $Π$-VS with respect to the number of allowed splits, showing para-NP-hardness for $K_3$-Free-VS and deriving an XP-algorithm when each vertex is only allowed to be split at most once.

On the Complexity of Establishing Hereditary Graph Properties via Vertex Splitting

TL;DR

This work analyzes the complexity of turning graphs into members of a hereditary property via vertex splits, formalized as -Vertex Splitting (-VS). It yields a tight dichotomy for properties defined by forbidden induced subgraphs with at most three vertices, and then develops a general reduction framework (via and admissible splitting configurations) to prove NP-hardness for broad families of forbidden subgraphs, including biconnected and higher-connected cases. The authors further show hardness for infinite families of cycles (e.g., Bipartite-VS and Perfect-VS) and examine the parameterized complexity with respect to the number of splits , obtaining para-NP-hardness in some cases and XP algorithms when each vertex can be split at most once. Overall, the paper maps the tractability frontier for -VS, highlighting the role of subgraph connectivity and cycle structure, and it outlines several open problems on the remaining tractable cases (e.g., and ).

Abstract

Vertex splitting is a graph operation that replaces a vertex with two nonadjacent new vertices and makes each neighbor of adjacent with one or both of the introduced vertices. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we explore the computational complexity of achieving a given graph property by a limited number of vertex splits, formalized as the problem Vertex Splitting (-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of -VS for graph properties characterized by forbidden subgraphs of size at most 3. Second, we provide a framework that allows to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Leveraging this framework we show NP-completeness when is characterized by forbidden subgraphs that are sufficiently well connected. In particular, we show that -Free-VS is NP-complete for each biconnected graph . Third, we study infinite families of forbidden subgraphs, obtaining NP-hardness for Bipartite-VS and Perfect-VS. Finally, we touch upon the parameterized complexity of -VS with respect to the number of allowed splits, showing para-NP-hardness for -Free-VS and deriving an XP-algorithm when each vertex is only allowed to be split at most once.
Paper Structure (29 sections, 37 theorems, 26 equations, 15 figures, 2 tables, 2 algorithms)

This paper contains 29 sections, 37 theorems, 26 equations, 15 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Graph Notation
  4. Vertex Splitting
  5. Embeddings and Hereditary Graph Properties
  6. Miscellaneous
  7. Properties Characterized by Small Forbidden Induced Subgraphs: Outline
  8. Properties Characterized by Small Forbidden Induced Subgraphs: Full Proof
  9. Indestructible Forbidden Induced Subgraphs
  10. Cluster Graphs With Clusters of at Most Two Vertices as Forbidden Induced Subgraphs
  11. Forbidden Induced Subgraphs That When Destroyed Introduce a Forbidden Induced P3-complement
  12. Properties With a Forbidden Induced Subgraph of Size at Most Two
  13. Two Polynomial-Time Solvable Restrictions of Cluster Vertex Splitting
  14. Cluster graphs with clusters of at most two vertices
  15. Cluster Graphs of at Most Two Clusters
  16. ...and 14 more sections

Key Result

Lemma 2.2

Let $G = (V,E)$ be a graph, and let $I \coloneqq \{v \in V \mid d_G(v) = 0\}$. Then, there are at most $k$ vertex splits that turn $G$ into a cluster graph if and only if $G$ admits a sigma clique cover with weight at most $|V| - |I| + k$.

Figures (15)

  • Figure 1: The graphs $K_3$ and $P_3$ are split to break isomorphism. An induced embedding of $\overline{P_3}$ is highlighted in each graph after the split.
  • Figure 2: A graph $G$ for which $(G, 2)$ is a positive instance of $\mathop{\mathrm{Free}}\nolimits_\prec( \left \{ P_3, \overline{K_3} \right \} )$-Vertex Splitting. A sigma clique cover of size two and weight ten is highlighted in gray. One of the $3 \cdot 2 \cdot 3 = 18$ induced $P_3$'s is marked in red. The set of midpoints for all induced $P_3$'s is given by the intersection of the two cliques. Note that this set induces a clique of size two.
  • Figure 3: Illustration used in the proof of \ref{['lemma:p3_midpoints_induce_scc']}. The dashed line indicates a non-edge.
  • Figure 4: Illustration of all possible replacements performed in both directions of the proof of \ref{['lemma:subdivided_vc_reduction']}. The coloring indicates which vertices are part of the respective vertex cover.
  • Figure 5: Example of \ref{['definition:constr']}. The construction is carried out for the "skeleton" graph $\vec{G}$, a splitting configuration $C$, and the set of vertices $S$ marked in blue. The edge gadget graph is $H$; its "$a$-end" is $u_2$ and its "$b$-end" is $u_3$. The subscript of $\chi$, $\mathop{\mathrm{Constr}}\nolimits(\vec{G}, C, S)$, is dropped for brevity.
  • ...and 10 more figures

Theorems & Definitions (79)

  • Definition 2.1
  • Lemma 2.2: Firbas et al. firbas_cluster_2023, Lemma 4.3
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 69 more