Geodesic
On Independent Domination in Planar Cubic Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 4, pp. 841-853

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A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number, i(G), of G is the minimum cardinality of an independent dominating set. Goddard and Henning [Discrete Math. 313 (2013) 839–854] posed the conjecture that if G ∉{ K_3,3, C_5 □ K_2 } is a connected, cubic graph on n vertices, then i(G) ≤ 3/8 n, where C_5 □ K_2 is the 5-prism. As an application of known result, we observe that this conjecture is true when G is 2-connected and planar, and we provide an infinite family of such graphs that achieve the bound. We conjecture that if G is a bipartite, planar, cubic graph of order n, then i(G) ≤ 1/3 n, and we provide an infinite family of such graphs that achieve this bound.
Keywords: independent domination number, domination number, cubic graphs 
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Abrishami, Gholamreza; Henning, Michael A.; Rahbarnia, Freydoon. On Independent Domination in Planar Cubic Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 4, pp. 841-853. http://geodesic.mathdoc.fr/item/DMGT_2019_39_4_a5/