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...

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S . A paired-dominating set of G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired-dominating set) is the total domination number γt(G) (respectively, the paired-domination number γpr(G) ). We giv...

This paper compares asymptotic “average case”performance of two closely related algorithms for finding small connected dominating sets. The stochastic model is that instances are random unit ball graphs formed from n random points in an `n × `n × `n cube. The first algorithm, widely known as “Rule 1”, is proved to be ineffective asymptotically: if `n = O( 3 √ n log n ), then with asymptotic pro...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...