γ k ( n ) = max { n / ( 2 k + 1 ) , 1 } for Maximal Outerplanar Graphs with n mod ( 2 k + 1 ) ≤ 6 L iang

Let G = (V, E) be an undirected graph with a set V of nodes and a set E of edges, |V | = n. A node v is said to distance-k dominate a node w if w is reachable from v by a path consisting of at most k edges. A set D ⊆ V is said a distance-k dominating set if every node can be distance-k dominated by some v ∈ D. The size of a minimum distance-k dominating set, denoted by γk(G), is called the distance-k domination number of G. The value γk(n) is defined by γk(n) = max{γk(G) : G has n nodes}. This paper considers γk(n) for maximal outerplanar graphs. There is a conjecture γk(n) = max{ n/(2k + 1) , 1}, which was proved for k = 1, 2. This paper gives a unified and simpler proof for k = 1, 2, 3. In fact, a stronger result is shown that for all n > 2k and r = n mod (2k + 1) ≤ 6, there exist at least 2k + 1 − r distinct distance-k dominating sets of size at most n/(2k + 1) , which can be found in linear time.