Let G = (V,E) be a simple graph. A subset Dof V (G) is a (k, r)dominating set if for every vertexv ∈ V −D, there exists at least k vertices in D which are at a distance utmost r from v in [1]. The minimum cardinality of a (k, r)-dominating set of G is called the (k, r)-domination number of G and is denoted by γ(k,r)(G). In this paper, minimal (k, r)dominating sets are characterized. It is prove...

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-d...

For simple graphs G and H, let f(G, H) denote the least integer N such that every coloring of the edges of KN contains either a monochromatic copy of G or a rainbow copy of H. Here we investigate f(G, H) when H = Pk. We show that even if the number of colors is unrestricted when defining f(G, H), the function f(G, Pk), for k = 4 and 5, equals the (k − 2)– coloring diagonal Ramsey number of G.

For a graph G, a positive integer k, k ≥ 2, and a non-negative integer with z < k and z 6= 1, a subset D of the vertex set V (G) is said to be a non-z (mod k) dominating set if D is a dominating set and for all x ∈ V (G), |N [x] ∩ D| 6≡ z (mod k). For the case k = 2 and z = 0, it has been shown that these sets exist for all graphs. The problem for k ≥ 3 is unknown (the existence for even values...

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...

We propose a fast, silent self-stabilizing protocol building a distance-k independent dominating set, named FID. The convergence of the protocol FID is established for any computation under the unfair distributed scheduler. The protocol FID reaches a terminal (also legitimate) configuration in at most 4n+k rounds, where n is the network size; it requires (k + 1)log(n + 1) bits per node. keyword...

A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P . In 1988 H. Broersma stated a result implying that every n-vertex kconnected graph G such that σ(k+2)(G) ≥ n− 2k − 1 contains a dominating path. We show that every n-vertex k-connected graph with σ2(G) ≥ 2n k+2 + f(k) contains a dominating path of length at most O(|T |), where T is a minimum do...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

This study aims at exploring postcolonial themes raised by Andrea Hirata’s Laskar Pelangi (Rainbow Warriors). Specifically, it will reveal the characteristics of hybridity found in the novel that prove this literary work may be categorized as postcolonial writing despite the fact that western or white colonialism has no impact or trace at all in the novel. Furthermore, the study will prove that...

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...