On graphs with no induced subdivision of $K_4$

TL;DR

This paper characterizes the structure of graphs excluding certain subdivisions and induced subgraphs, leading to a polynomial-time recognition algorithm and a proof that such graphs are 3-colorable.

Contribution

It provides a new decomposition and structure theorem for graphs without induced subdivisions of $K_4$ and wheels, and proves their 3-colorability.

Findings

Graphs without induced subdivision of $K_4$ have a specific structure.

Graphs in class $ ext{C}$ are 3-colorable.

Recognition of graphs in $ ext{C}$ is polynomial-time feasible.

Abstract

We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.