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Keywords:

  • dominating set;
  • triangulation;
  • projective plane;
  • torus;
  • Klein bottle;
  • representativity

Abstract

Let G be a graph and let SV(G). We say that S is dominating in G if each vertex of G is in S or adjacent to a vertex in S. We show that every triangulation on the torus and the Klein bottle with n vertices has a dominating set of cardinality at most equation image. Moreover, we show that the same conclusion holds for a triangulation on any non-spherical surface with sufficiently large representativity. These results generalize that for plane triangulations proved by Matheson and Tarjan (European J Combin 17 (1996), 565–568), and solve a conjecture by Plummer (Private Communication). © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 17–30, 2010