On a theorem of Sewell and Trotter

Sewell and Trotter proved that every connected α -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.