Bounding the size of equimatchable graphs of fixed genus (Q1043800)

A graph \(G\) is equimatchable if every matching in \(G\) extends to a maximum matching in \(G\). In [Discrete Math. 266, No.1--3, 263--274 (2003; Zbl 1022.05065)], \textit{K.~Kawarabayashi}, \textit{M. D. Plummer} and \textit{A. Saito} showed that there are only a finite number of 3-connected equimatchable planar graphs. The present paper shows that in a surface (orientable or non-orientable) of any fixed genus there are only a finite number of 3-connected equimatchable graphs having a minimal imbedding of representativity at least three. (The latter hypothesis can be dropped, for non-bipartite graphs.) The proof uses the Gallai-Edmonds decomposition theorem for matchings.