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Journal Articles Journal of Combinatorial Theory, Series B Year : 2012

On graphs with no induced subdivision of K4

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.

Dates and versions

lirmm-00736503 , version 1 (28-09-2012)

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Benjamin Lévêque, Frédéric Maffray, Nicolas Trotignon. On graphs with no induced subdivision of K4. Journal of Combinatorial Theory, Series B, 2012, 102, pp.924-947. ⟨10.1016/j.jctb.2012.04.005⟩. ⟨lirmm-00736503⟩
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