Domination of maximal K4 K 4 -minor free graphs and maximal K2,3 K 2 , 3 -minor free graphs, and disproofs of two conjectures on planar graphs ?
It is well known that a graph is outerplanar if and only if it is K4 K 4 -minor free and K2,3 K 2 , 3 -minor free. Campos and Wakabayashi (2013) recently proved that ? ( G ) ? ? n + k 4 ? for any maximal outerplanar graph G G of order n?3 n ? 3 with k k vertices of degree 2, where ?(G) ? ( G ) denotes the domination number of G G . Tokunaga (2013) provided a short proof for the above theorem. Based on some structural properties of K2,3 K 2 , 3 -minor free graphs and K4 K 4 -minor free graphs, applying the idea of Tokunaga we extend the theorem of Campos and Wakabayashi to all maximal K4 K 4 -minor free graphs and all maximal K2,3 K 2 , 3 -minor free graphs. We also disprove two conjectures of Tokunaga on planar graphs.
3 outerplanar graph domination number k4 k 4 -minor free graph k4 k 4 k4 k 4 k 4 k2 3 k 2 3 -minor free graph k2 3 k2 triangulation