Clique-width and induced topological minors

2026-05-14

AI Summary

This study resolves an open problem posed by Dabrowski et al. concerning the characterization of graphs \( H \) for which the class of graphs excluding induced subdivisions of \( H \) has bounded clique-width. By integrating techniques from induced subgraph analysis, clique-width theory, and structural properties of subdivisions, the authors establish that this boundedness property holds if and only if \( H \) is an induced subgraph of the path \( P_4 \), the claw, or the diamond. This result provides a complete characterization of the equivalence between forbidden induced subdivisions and bounded clique-width, thereby yielding an exact classification theorem that precisely delineates the boundary between bounded and unbounded clique-width within this context.

Abstract

A $P_4$ is a chordless path on four vertices. A diamond is a graph obtained from a clique of size four by removing one edge of the clique. A paw is a graph obtained from a clique of size four by removing two adjacent edges of the clique. We prove that for a graph $H$, the class of graphs with no induced subdivision of $H$ has bounded clique-width if and only if $H$ is an induced subgraph of $P_4$, the paw, or the diamond. This answers a~question of Dabrowski, Johnson, and Paulusma.

Problem

• clique-width
• induced topological minor
• bounded clique-width
• P4
• diamond

Innovation

• clique-width
• induced topological minor
• bounded graph class
• P4
• graph characterization