A set S of vertices in graph [Formula: see text] is a [Formula: see text], abbreviated TRDS, of G if every vertex of G is adjacent to a vertex in S and every vertex of [Formula: see text] is adjacent to a vertex in [Formula: see text]. The [Formula: see text] of G, denoted by [Formula: see text], is the minimum cardinality of a TRDS of G. Jiang and Kang (J Comb Optim. 19:60-68, 2010) characteri...

For a graph G = (V,E), a set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S as well as another vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper we find all graphs G satisfying γr(G) = n− 3, where n is the order of G.

the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k , the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a line...

Let G = (V, E) be a graph and let S ⊆ V . The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), i...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear...