We propose a memory efficient self-stabilizing protocol building distance-k independent dominating sets. A distance-k independent dominating set is a distance-k independent set and a distance-k dominating set. Our algorithm, named SID, is silent; it converges under the unfair distributed scheduler (the weakest scheduling assumption). The protocol SID is memory efficient : it requires only log(2...

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...

We study a pair of NP-hard problems aimed at finding small sets of faces in planar graphs. In the disk dimension problem, we are given a planar graph G, and seek the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with bounded disk dimension are derived. We show how to obtain an outerplanar subgraph o...

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...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ j for each x...

‎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 $G$...