Given a graph G, a dominating set D is a set of vertices such that any vertex in G has at least one neighbor (or possibly itself) in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) and properties ...

‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over ...

Let $D$ be a finite and simple digraph with vertex set $V(D)$. A weak signed Roman dominating function (WSRDF) on is $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the condition that $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consists of $v$ allvertices from which arcs go into $v$. The weight WSRDF $f$ $\sum_{v\in V(D)}f(v)$. domination number $\gamma_{wsR}(D)$ minimum $D$. In...

A signed tree-coloring of a signed graph (G, σ) is a vertex coloring c so that G(i,±) is a forest for every i ∈ c(u) and u ∈ V(G), where G(i,±) is the subgraph of (G, σ) whose vertex set is the set of vertices colored by i or −i and edge set is the set of positive edges with two end-vertices colored both by i or both by −i, along with the set of negative edges with one end-vertex colored by i a...

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ n), where c = 36 √ 34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O( √ γ(G)), and that such a tree decomposition can be found in O( √ γ(G)n) time. The same technique can be used to show that the k-face cover probl...

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 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...