γk(n) = max {⌊n/(2k+1)⌋, 1} for Maximal Outerplanar Graphs with n mod (2k+1) ≤ 6

Abstract

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 dominatea 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 setif every node can be distance-k dominated by some v ∈ D. The size of a minimum distance-k dominating set, denoted by γ_k(G), is called the distance-k domination number of G. The value γ_k(n) is defined by γ_k(n) = max{γ_k(G) : G has n nodes}. This paper considers γ_k(n) for maximal outerplanar graphs. There is a conjecture γ_k(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.