Vertices contained in all or in no minimum k - dominating sets of a tree ∗
نویسندگان
چکیده
Let k be a positive integer and G = (V,E) be a simple graph. A subset S ⊆ V is dominating in G, if for each vertex v ∈ V \ S, N(v) ∩ S 6= ∅. In 1985, Fink and Jacobson gave a generalization of the concept of dominating sets in graphs. A subset S of V is kdominating in G, if every vertex of V \ S is adjacent to at least k vertices in S. In this paper, we characterize vertices that are in all or in no minimum k-dominating sets in a tree for k ≥ 2.
منابع مشابه
k-TUPLE DOMATIC IN GRAPHS
For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...
متن کامل$k$-tuple total restrained domination/domatic in graphs
For any integer $kgeq 1$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple total dominating set of $G$ if any vertex of $G$ is adjacent to at least $k$ vertices in $S$, and any vertex of $V-S$ is adjacent to at least $k$ vertices in $V-S$. The minimum number of vertices of such a set in $G$ we call the $k$-tuple total restrained domination number of $G$. The maximum num...
متن کاملVertices Contained in all or in no Minimum Paired-Dominating Set of a Tree
A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree.
متن کاملStrength of strongest dominating sets in fuzzy graphs
A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...
متن کاملVertices Belonging to All or to No Minimum Locating Dominating Sets of Trees
A set D of vertices in a graph G is a locating-dominating set if for every two vertices u, v of G \ D the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. In this paper, we characterize vertices that are in all or in no minimum locating dominating sets in trees. The characterization guarantees that the γL-excellent tree can be recognized in a polynomial time.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014