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

Excluding four-edge paths and their complements