Clique-width and induced topological minors

TL;DR

This paper characterizes which graphs have classes of graphs with bounded clique-width when excluding induced subdivisions, specifically identifying $P_4$, paw, and diamond as key structures.

Contribution

It provides a complete characterization of graphs $H$ for which the class of graphs with no induced subdivision of $H$ has bounded clique-width.

Findings

Bounded clique-width classes are characterized by $H$ being an induced subgraph of $P_4$, paw, or diamond.

Answers a question posed by Dabrowski, Johnson, and Paulusma.

Establishes a precise structural criterion for bounded clique-width in induced subdivision classes.

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.