An outer-independent double Roman dominating function (OIDRDF) of a graph G is h:V(G)→{0,1,2,3}</...

Given graph G = (V,E), a dominating set S is a subset of vertex set V such that any vertex not in S is adjacent to at least one vertex in S. The domination number of a graph G is the minimum size of the dominating sets of G. In this paper we study some results on domination number, connected, independent, total and restrained domination number denoted by γ(G), γc(G) ,γi(G), γt(G) and γr(G) resp...

*Supported partly by the National Science Council of the Republic of China under grant NSC82-0208-M009-050. IDIMACS permanent member. lCurrent address: Laboratory for Computer Science, MIT, Cambridge, MA 02139. Emaih [email protected].

Let H be a digraph possibly with loops and let D be a digraph whose arcs are colored with the vertices of H (an H-colored digraph). A walk (path) P in D will be called an H-restricted walk (path) if the colors displayed on the arcs of P form a walk in H. An H-restricted kernel N is a set of vertices of D such that for every two different vertices in N there is no H-restricted path in D joining ...

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 ...

The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixe...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. Introduction. This note is motivated by the following...

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, the sums and products of ψ(G1) and ψ(G2) are examined where G1 ⊕ G2 = K(s, s), and ψ is the independence, domination, or independent domination number, inter alia. In particular, it is shown that the maximum valu...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the notion of k-dominating ...