Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number k(G), the connected k-domination number c k (G); the k-independent domination number i k (G) and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then irk(G) = k(G) = k(G), and that for k ...

We consider families of analytic functions with Taylor coefficients-polynomials in the parameter λ: fλ(z) = ∑∞ k=0 ak(λ)z k, ak ∈ C[λ]. Let R(λ) be the radius of convergence of fλ. The “Taylor domination” property for this family is the inequality of the following form: for certain fixed N and C and for each k ≥ N + 1 and λ, |ak(λ)|R(λ) ≤ C max i=0,...,N |ai(λ)|R(λ). Taylor domination property ...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one t...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...

‎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...

4 For a graph G = (V,E), a set S ⊆ V is a dominating set if every vertex in 5 V is either in S or is adjacent to a vertex in S. The domination number γ(G) 6 of G is the minimum order of a dominating set in G. A graph G is said to be 7 domination vertex critical, if γ(G− v) < γ(G) for any vertex v in G. A graph 8 G is domination edge critical, if γ(G∪ e) < γ(G) for any edge e / ∈ E(G). We 9 call...

The domination game is played on an arbitrary graph $G$ by two players, Dominator and Staller. It is known that verifying whether the game domination number of a graph is bounded by a given integer $k$ is PSPACE-complete. On the other hand, it is showed in this paper that the problem can be solved for a graph $G$ in $mathcal O(Delta(G)cdot |V(G)|^k)$ time. In the special case when $k=3$ and the...

For a simple graph G, the independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds for the independent domination number of K1,k+1-free graphs, as functions of the order, size and k. Also we present a lower bound for the size of connected graphs with given order and value of independent domination ...