On a Theorem of Sewell and Trotter

TL;DR

This paper provides a simpler proof of Sewell and Trotter's 1993 theorem, which characterizes connected alpha-critical graphs without certain subdivisions and relates to stable sets.

Contribution

It offers a more straightforward proof of a key theorem in graph theory concerning alpha-critical graphs and their subdivisions.

Findings

Simplified proof of Sewell and Trotter's theorem

Clarification of the structure of alpha-critical graphs

Implications for stable set min-max relations

Abstract

Sewell and Trotter [J. Combin. Theory Ser. B, 1993] proved that every connected alpha-critical graph that is not isomorphic to K_1, K_2 or an odd cycle contains a totally odd K_4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K_4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter's theorem.