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arXiv · 1606.07662

Reducing quadrangulations of the sphere and the projective plane

Elke Fuchs·Laura Gellert
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Abstract

We show that every quadrangulation of the sphere can be transformed into a $4$-cycle by deletions of degree-$2$ vertices and by $t$-contractions at degree-$3$ vertices. A $t$-contraction simultaneously contracts all incident edges at a vertex with stable neighbourhood. The operation is mainly used in the field of $t$-perfect graphs. We further show that a non-bipartite quadrangulation of the projective plane can be transformed into an odd wheel by $t$-contractions and deletions of degree-$2$ vertices. We deduce that a quadrangulation of the projective plane is (strongly) $t$-perfect if and only if the graph is bipartite.

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Elke Fuchs, Laura Gellert. 2016-06-24. Reducing quadrangulations of the sphere and the projective plane. https://arxiv.org/abs/1606.07662

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