Excluding four-edge paths and their complements

Abstract

We prove that a graph G contains no induced four-edge path and no induced complement of a four-edge path if and only if G is obtained from five-cycles and split graphs by repeatedly applying the following operations: substitution, split graph unification, and split graph unification in the complement ("split graph unification" is a new class-preserving operation that is introduced in this paper).

28 pages, the paper arXiv:1410.0871 resulted from the merging of this manuscript together with 'On $(P_5,\overline{P_5})$-free graphs', by L. Esperet, L. Lemoine, and F. Maffray (2013)