@techreport{TR-IC-99-15,
  number = {IC-99-15},
  author = {Cândido F. X. de {Mendonça N.} and Érico F. {Xavier} and Jorge Stolfi and Luerbio Faria and Celina M. H. de {Figueiredo}},
  title = {The Vertex Deletion Number and Splitting Number of a Triangulation of {$C_n \times C_m$}},
  month = {June},
  year = {1999}, 
  institution = {Institute of Computing, University of Campinas},
  note = {In English, 8 pages.
    \par\selectlanguage{english}\textbf{Abstract}
      The vertex deletion number $\phi(G)$ of a graph $G$ is the
      minimum number of vertices that must be deleted from $G$ to
      produce a planar graph. The splitting number $\sigma(G)$ of $G$
      is the smallest number of vertex splitting operations that must
      be applied to $G$ to make it planar. Here we determine these
      topological invariants for the graph family ${\cal T}_{C_n
      \times C_m}$, a regular triangulation of the torus obtained by
      adding parallel diagonal edges to the faces of the rectangular
      toroidal grid $C_n \times C_m$. Specifically, we prove that the
      obvious upper bound $\phi = \sigma = \min\{n,m\}$ is also a
      lower bound.
  }
}