On graphs with no induced subdivision of K4

Authors:
Benjamin LéVêQue;FréDéRic Maffray;Nicolas Trotignon
Affiliations:
CNRS, LIRMM, 161 rue Ada, 34392 Montpellier Cedex 05, France;CNRS, Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble Cedex, France;CNRS, LIP, ENS Lyon, INRIA, Université de Lyon, France
Venue:
Journal of Combinatorial Theory Series B
Year:
2012

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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 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 C is 3-colorable and entails a polynomial-time recognition algorithm for membership in C. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.