Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph...

Let G be a connected graph of order n, and let NC2(G) denote min{[N(u)UN(v)[: dist(u, v )= 2}, where dist(u, v) is the distance between u and v in G. A cycle C in G is called a dominating cycle, if V(G)\V(C) is an independent set in G. In this paper, we prove that if G contains a dominating cycle and ~ ~> 2, then G contains a dominating cycle of length at least min{n,2NC2(G)3}. ~ 1997 Elsevier ...

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...

A distance-hereditary graph is a connected graph in which every induced path is isometric, i.e., the distance of any two vertices in an induced path equals their distance in the graph. We present a linear time labeling algorithm for the minimum cardinality connected r-dominating set and Steiner tree problems on distance-hereditary graphs. q 1998 John Wiley & Sons, Inc. Networks 31: 177–182, 1998

Let nand k be positive integers and let G be a graph. A set D of vertices of G is defined to be an (n, k )-dominating set of G if every vertex of V( G) D is within distance n from at least k vertices of D. The minimum cardinality among all (n, k )-dominating sets of G is called the (n, k )-domination number of G and is denoted by 'Yn,k(G). A set I of vertices of G is defined to be an (n, k)inde...

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

We present an overview of the recent progress of applying computational geometry techniques to solve some questions, such as topology construction and broadcasting, in wireless ad hoc networks. Treating each wireless device as a node in a two dimensional plane, we model the wireless networks by unit disk graphs in which two nodes are connected if their Euclidean distance is no more than one. We...

We propose algorithms for efficiently maintaining a constant-approximate minimum connected dominating set (MCDS) of a geometric graph under node insertions and deletions. Assuming that two nodes are adjacent in the graph iff they are within a fixed geometric distance, we show that an O(1)-approximate MCDS of a graph in R d with n nodes can be maintained with polylogarithmic (in n) work per node...

PROBLEM DEFINITION Consider a graph G = (V,E). A subset C of V is called a dominating set if every vertex is either in C or adjacent to a vertex in C. If, furthermore, the subgraph induced by C is connected, then C is called a connected dominating set. A connected dominating set with a minimum cardinality is called a minimum connected dominating set (MCDS). Computing a MCDS is an NP-hard proble...