Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to bebegin{itemize}item a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The R...

Using connected dominating set (CDS) to serve as a virtual backbone in a wireless networks can save energy and reduce interference. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. A k-connected m-fold dominating set ((k,m)-CDS) of a graph G is a node set D such that every node in V \ D has at least m neighbor...

A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph G is strongly rainbow-connected. The minimum number of colors needed to make G rainbow-connected is known as the rainbow connection number...

A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number...

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − →rc(T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of kth diameter 2 has rainbow k-co...

‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ is a hop dominating set‎‎if every vertex outside $S$ is at distance two from a vertex of‎‎$S$‎. ‎A hop dominating set $S$ which induces a connected subgraph‎ ‎is called a connected hop dominating set of $G$‎. ‎The‎‎connected hop domination number of $G$‎, ‎$ gamma_{ch}(G)$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of $G$...

In this paper, we consider counting the number of ways to place kings on an k × n chessboard, such that every square is dominated by a king. Let f(k, n) be the number of dominating configurations. We consider the asymptotic behavior of the function f(k, n).

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

A new class of rainbows is created when a droplet is illuminated from the inside by a point light source. The position of the rainbow depends on both the index of refraction of the droplet and the position of the light source, and the rainbow vanishes when the point source is too close to the center of the droplet. Here we experimentally measure the position of the transmission and one-internal...

Given a graph G, a k-dominating set of G is a subset S of its vertices with the property that every vertex of G is either in S or has at least k neighbors in S. We present a new incremental local algorithm to construct a k-dominating set. The algorithm constructs a monotone family of dominating sets D1 ⊆ D2 . . . ⊆ Di . . . ⊆ Dk such that each Di is an i-dominating set. For unit disk graphs, th...