The (theta, wheel)-free graphs Part I: only-prism and only-pyramid graphs

TL;DR

This paper establishes structural theorems for certain hereditary graph classes excluding specific Truemper configurations, leading to polynomial recognition algorithms and laying groundwork for future optimization problem solutions.

Contribution

It provides the first structure theorems for graphs excluding thetas, wheels, and either prisms or pyramids, enabling polynomial recognition algorithms for these classes.

Findings

Proved structure theorems for graphs without thetas, wheels, and prisms or pyramids.

Developed polynomial time recognition algorithms for these graph classes.

Set the stage for solving optimization problems on these classes in subsequent parts.

Abstract

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs.