Abstract

For k ≥ 1 an integer, a set S of vertices in a graph G with minimum degree at least k − 1 is a k-tuple dominating set of G if every vertex of S is adjacent to at least k − 1 vertices in S and every vertex of V (G) / S is adjacent to at least k vertices in S; that is, |NG[v] ∩ S| ≥ k for every vertex v of G where NG[v] denotes the closed neighborhood of v which consists of v and all neighbors of v. A k-tuple restrained dominating set of G is a k-tuple dominating set S of G with the additional property that every vertex outside S has at least k neighbors outside S. The mini- mum cardinality of a k-tuple restrained dominating set of G is the k-tuple restrained domination number of G. When k = 1, the k-tuple restrained domination number is the well-studied restrained domination number. In this paper, we determine the k-tuple restrained domination number of several classes of graphs. Tight bounds on the k-tuple restrained domination number of a general graph are established. We present basic properties of the k-tuple restrained domatic number of a graph which is the maximum number of the classes of a partition of V (G) into k-tuple restrained dominating sets of G.

摘要

对于ķ ≥1的整数，a组š在图中的顶点的ģ至少以最小程度ķ - 1是ķ元组支配集的ģ如果每个顶点小号是邻近至少ķ -在1个顶点小号和V ( G )/ S 的每个顶点至少与S 中的k个顶点相邻；也就是说，| NG [ v ] ∩ S | ≥ ķ为每个顶点v的ģ其中NG [ v ]表示的闭合附近v它由v和的所有邻居v。甲ķ元组约束的控制集ģ是ķ元组支配集小号的ģ与附加属性，该属性外的每个顶点小号具有至少ķ外部邻居小号。一的微型妈妈基数ķ元组约束支配的组G ^是ķ的元组约束控制数ģ。当k = 1 时，k元组约束支配数是经过充分研究的约束支配数。在本文中，我们确定了几类图的k元组约束支配数。建立了一般图的k元组约束支配数的严格界限。所述的本我们基本性质ķ元组抑制的曲线图，其是一个分区的的类别的最大数目的数domatic V（G ^成）ķ抑制支配套元组G ^。