Domination Of Maximal K₄-minor Free Graphs And Maximal 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-minor free and K2,3-minor free. Campos and Wakabayashi (Discrete Appl. Math.161 (2013) 330-335) recently proved that γ(G)≤[4/n+k]for any maximal outerplanar graph G of order n≥ 3 with k vertices of degree 2, where 7(G) denotes the domination number of G. Tokunaga (Discrete Appl. Math.161 (2013) 3097-3099) provided a short proof for the above theorem and posed five conjectures. Based on some structural properties of K2,3-minor free graphs and K4-minor free graphs in Section 2, applying the idea of Tokunaga we extend he theorem of Campos and Wakabayashi to all maximal K4-minor free graphs and all maximal K2,3-minor free graphs, and we prove it in Section 3. Based on some related conjectures and results, we conjecture that γ(G)≤3/n for any 2-connected 3-regular planar graph G of order n. We also disprove two conjectures of Tokunaga on planar graphs.