Unique minimum domination in trees

A set D of vertices in a graph G is a distance-k dominating set if every vertex of G either is in D or is within distance k of at least one vertex in D. A distance-k dominating set of G of minimum cardinality is called a minimum distance-k dominating set of G. For any graph G and for a subset F of the edge set of G the set F is an edge dominating set of G if every edge of G either is in D or is adjacent to at least one edge in D. An edge dominating set of G of minimum cardinality is called a minimum edge dominating set of G. We characterize trees with unique minimum distance-k dominating sets, which is a generalization of a result of Gunther, Hartnell, Markus, and Rall. Further, we give a characterization of trees with unique minimum edge dominating sets, which contains some results of Topp. 1 Terminology and Introduction For any graph G the vertex set and the edge set of G are denoted by V (G) and E(G), and n(G) = |V (G)| and m(G) = |E(G)|. The number of components of G is denoted by κ(G). For any subset A ⊆ V (G) we define the induced subgraph G[A] as the graph with vertex set A and edge set {ab ∈ E(G) | a, b ∈ A}. For any set A ⊆ V (G) and any vertex x ∈ V (G) we define G − A = G[V (G) \ A] and G − x = G − {x}. For two vertices x and y in a connected graph G the distance d(x, y) between x and y is the minimum number of edges of a path in G from x to y. If we define e(v) = maxw∈V (G) d(v, w), then the diameter of G is diam(G) = maxv∈V (G) e(v) and the radius of G is rad(G) = minv∈V (G) e(v). For any vertex x ∈ V (G) the open k-neighborhood of x, denoted Nk(x), is the set Nk(x) = {y ∈ V (G) | y = x and d(x, y) ≤ k} and the set Nk[x] = Nk(x) ∪ {x} is called the closed k-neighborhood of x. If A ⊆ V (G), then Nk(A) = ⋃x∈A Nk(x) and Nk[A] = Nk(A) ∪ A. For a subset D of V (G) and a vertex x ∈ D the set Pk(x,D) = Nk[x] \Nk[D \ {x}] is called the private k-neighborhood of x with regard to D and a vertex y ∈ Pk(x,D) is called a private k-neighbor of x with regard to D. Australasian Journal of Combinatorics 25(2002), pp.117–124 A set D ⊆ V (G) is a distance-k dominating set of G if Nk[D] = V (G). The minimum cardinality of a distance-k dominating set is called the distance-k domination number denoted by γ≤k(G). A distance-k dominating set D of G with cardinality γ≤k(G) is called a γ≤k-set or a minimum distance-k dominating set. Note that the case k = 1 leads to the ordinary domination. There are several publications on distance domination as e.g. [1], [2], [13], [14] and the chapter ‘Distance domination in graphs’ by M.A. Henning in [11]. For any subset B ⊆ E(G) we define the subgraph G(B) as the graph with edge set B and vertex set {v, w ∈ V (G) | vw ∈ B}. For any set B ⊆ E(G) and any edge e ∈ E(G) we define G − B = G(E(G) \ B) and G − e = G − {e}. Notice that the subgraphs G−B and G− e contain no isolated vertices. A subset F of the edge set E(G) is an edge dominating set of G if every edge in G either is in F or is adjacent to at least one edge in F . The edge domination number γ′(G) is the smallest cardinality of all edge dominating sets and an edge dominating set of cardinality γ′(G) is called a minimum edge dominating set of G. The edge domination is studied in numerous publications as e.g. in [3], [4], [12], [15], and in [18]. For other graph theory terminology we follow [10]. 2 Unique minimum distance domination in trees Theorem 2.1 Let T be a tree of order at least 3, let D be a subset of V (T ), and let k be a positive integer. Then the following conditions are equivalent: (i) D is the unique γ≤k-set of T . (ii) D is a distance-k dominating set of T such that every vertex in D has at least two private k-neighbors v and w with d(v, w) = 2k. (iii) D is a γ≤k-set of T such that γ≤k(T − x) > γ≤k(T ) for every vertex x ∈ D. Proof. (i) ⇒ (ii): Let D be the unique γ≤k-set of T . Then, we have |Pk(x,D)| ≥ 2 for every vertex x ∈ D. Suppose there is a vertex x ∈ D such that d(a, b) < 2k for every pair of vertices in Pk(x,D). If d(a, x) < k for every vertex a in Pk(x,D), then for some arbitrary, fixed z ∈ N1(x) we have d(a, z) ≤ k for every vertex a in Pk(x,D), and (D \ {x}) ∪ {z} is a γ≤k-set of T different from D, which is a contradiction. Hence, there is a vertex a ∈ Pk(x,D) with d(x, a) = k. Let z ∈ N1(x) with d(z, a) = k − 1. Suppose there is a vertex b ∈ Pk(x,D) with d(z, b) > k. Then d(x, b) = k and the vertex x lies on the unique path from a to b. This yields the contradiction d(a, b) = d(a, x) + d(x, b) = 2k. Therefore d(z, b) ≤ k for every b ∈ Pk(x,D) and (D \ {x}) ∪ {z} is a γ≤k-set of T different from D, which again is a contradiction. (ii) ⇒ (i): We prove this by induction on the order n(T ). If a tree T has a distancek dominating set D as in (ii), then the diameter of T is greater or equal 2k and n(T ) ≥ 2k + 1. First, let T be a tree of order n(T ) = 2k + 1, that has a distance-k dominating set D as in (ii). Since the diameter of T is greater or equal 2k, the tree T