Graphs with no induced five-vertex path or antipath

TL;DR

This paper characterizes graphs that exclude induced five-vertex paths and their complements, showing they can be constructed from basic graphs using specific operations including a new one called split unification.

Contribution

It introduces a novel graph operation called split unification and provides a complete structural characterization of graphs avoiding induced five-vertex paths and their complements.

Findings

Graphs with no induced 5-path or its complement are built from 5-cycles and split graphs.

The paper introduces and formalizes the split unification operation.

Structural characterization enables better understanding of these graph classes.

Abstract

We prove that a graph $G$ contains no induced $5$-vertex path and no induced complement of a $5$-vertex path if and only if $G$ is obtained from $5$-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class-preserving operation introduced here.