Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...

Let $D$ be a finite and simple digraph with vertex set $V(D)$‎.‎A signed total Roman $k$-dominating function (STR$k$DF) on‎‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

4 In this paper, we study the domination number, the global dom5 ination number, the cographic domination number, the global co6 graphic domination number and the independent domination number 7 of all the graph products which are non-complete extended p-sums 8 (NEPS) of two graphs. 9

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed le sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimumdominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies dominat...

A dominating set in a graph G is a set S of vertices such that every vertex outside S has a neighbor in S; the domination number γ(G) is the minimum size of such a set. The independent domination number, written i(G), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured that if G is a connected cubic graph with sufficiently many vertices, then i(G)...

We let γ(G) and i(G) denote the domination number and the independent domination number ofG, respectively. Recently, Rad and Volkmann conjectured that i(G)/γ(G) ≤ ∆(G)/2 for every graph G, where ∆(G) is the maximum degree of G. In this note, we construct counterexamples of the conjecture for ∆(G) ≥ 6, and give a sharp upper bound of the ratio i(G)/γ(G) by using the maximum degree of G.