A subset S of nodes in a graph G is k-connected m-dominating set ((k , m)-cds) if the subgraph G[S] induced by and every $$v \in V {\setminus } S$$ has at least m neighbors S. In k -Connected -Dominating Set m)-CDS) problem, goal to find minimum weight (k, m)-cds node-weighted graph. For $$m \ge k$$ we obtain following approximation ratios. unit disk graphs improve ratio $$O(k \ln k)$$ Nutov (I...

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. ...

In a graph G = (V, E), a set D ⊂ V is a weak convex dominating(WCD) set if each vertex of V-D is adjacent to at least one vertex in D and d < D > (u, v) = d G (u, v) for any two vertices u, v in D. A weak convex dominating set D, whose induced graph < D > has no cycle is called acyclic weak convex dominating(AWCD) set. The domination number γ ac (G) is the smallest order of a acyclic weak conve...

‎for any integer $kgeq 1$‎, ‎a set $s$ of vertices in a graph $g=(v,e)$ is a $k$-‎tuple total dominating set of $g$ if any vertex‎ ‎of $g$ is adjacent to at least $k$ vertices in $s$‎, ‎and any vertex‎ ‎of $v-s$ is adjacent to at least $k$ vertices in $v-s$‎. ‎the minimum number of vertices of such a set‎ ‎in $g$ we call the $k$-tuple total restrained domination number of $g$‎. ‎the maximum num...

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

Let G be an IFG. Then V D  is said to bae a strong (weak) dominating set if every D V v   is strongly (weakly) dominated by some vertex in D. We denote the strong (weak) intuitionistic fuzzy dominating set by sid-set (wid-set). The minimum vertex cardinality over all the sid-set (wid-set) is called the strong (weak) dominating number of an IFG and is denoted by )] ( [ ) ( G G wid sid   In ...