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arXiv · 1309.1926

On graphs with no induced subdivision of $K_4$

Benjamin Lévêque·Frédéric Maffray·Nicolas Trotignon
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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.

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Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. 2013-09-08. On graphs with no induced subdivision of $K_4$. https://doi.org/10.1016/j.jctb.2012.04.005

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