Authors:
G. Abrishami, M.A. Henning, F. Rahbarnia
Title:
On independent domination in planar, cubic graphs
Source:
Discussiones Mathematicae Graph Theory
Received 02.08.2017, Revised 01.12.2017, Accepted 01.12.2017, doi: 10.7151/dmgt.2105

Abstract:
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 ∉ {K3,3, C5 \, \Box \, K2} is a connected, cubic graph on n vertices, then i(G) \le \frac{3}{8}n, where C5 \, \Box \, K2 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) \le \frac{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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