Cographs and 1-sums

TL;DR

This paper introduces sesquicographs, a new class of graphs generated from a single vertex using joins, 0-sums, and 1-sums, and explores their properties and relation to cographs.

Contribution

It defines sesquicographs, proves their closure under induced minors, and characterizes non-sesquicographs similar to cograph characterization.

Findings

Sesquicographs are closed under induced minors.

Cographs are characterized by the absence of a 4-vertex path.

An analogue characterization for sesquicographs is provided.

Abstract

A graph that can be generated from $K_1$ using joins and 0-sums is called a cograph. We define a sesquicograph to be a graph that can be generated from $K_1$ using joins, 0-sums, and 1-sums. We show that, like cographs, sesquicographs are closed under induced minors. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We obtain an analogue of this result for sesquicographs, that is, we find those non-sesquicographs for which every proper induced subgraph is a sesquicograph.