Global Domination in Planar Graphs

For any graph \( G = (V, E) \), \( D \subseteq V \) is a global dominating set if \( D \) dominates both \( G \) and its complement \( \overline{G} \). The global domination number \( \gamma_g(G) \) of a graph \( G \) is the fewest number of vertices required of a global dominating set. In general,

\[
\max\{\gamma(G), \gamma(\overline{G})\} \leq \gamma_g(G) \leq \gamma(G) + \gamma(\overline{G}),
\]

where \( \gamma(G) \) and \( \gamma(\overline{G}) \) are the respective domination numbers of \( G \) and \( \overline{G} \). We show that when \( G \) is a planar graph,

\[
\gamma_g(G) \leq \max\{\gamma(G) + 1, 4\}.
\]