On the complexity of edge subdivision to $H$-free graphs

2026-04-27

AI Summary

This study addresses the problem of determining whether a graph can be transformed, via at most $k$ edge subdivision operations, into a graph that contains no induced subgraph isomorphic to a fixed graph $H$ (the $H$-free Subdivision problem). Through a combination of structural graph analysis, parameterized algorithm design, and fine-grained complexity reductions based on the Exponential Time Hypothesis (ETH), the work provides the first comprehensive characterization of the problem’s computational complexity. It establishes that when $H$ satisfies certain structural conditions, the problem is NP-complete and admits no algorithm running in time $2^{o(k)}n^{O(1)}$, unless ETH fails. Otherwise, the problem is solvable in polynomial time, and a search-tree algorithm with runtime $2^{O(k)}n^{O(1)}$ is provided.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$ in $G$ by at most $k$ edge subdivisions. We show that the problem is polynomial-time solvable when every component of $H$ is a subdivided star or a subdivided bistar, and at most one component is a subdivided bistar. On the other hand, we prove that \textsc{$H$-free Subdivision} is NP-complete and, assuming the Exponential Time Hypothesis, admits no $2^{o(k)} n^{O(1)}$-time algorithm whenever $H$ satisfies any of the following conditions: \begin{itemize} \item $H$ has minimum degree at least $2$, and the neighborhood of every degree-$2$ vertex induces a $K_2$; \item the vertices of degree at least $3$ in $H$ induce a graph with at least two edges; \item $H$ has a triangle with two vertices of degree at least $3$; \item $H$ contains, as an induced subgraph, the graph obtained from two vertex-disjoint triangles by adding one edge between them; \item $H$ contains exactly one triangle; \item $H$ has girth at least $4$; \item $H$ is a tree with exactly two vertices of degree at least $3$ at distance $2$ or at least $4$. \end{itemize} A simple bounded search-tree algorithm for the problem runs in $2^{O(k)} n^{O(1)}$ time. Thus, for all hardness cases above, this running time is essentially optimal under ETH.

Problem

• edge subdivision
• H-free graphs
• induced subgraph
• NP-completeness
• parameterized complexity

Innovation

• edge subdivision
• H-free graphs
• parameterized complexity
• NP-completeness
• Exponential Time Hypothesis