The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  set of G. A dominating set D1  VD is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. ...

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 ...

Let G be a simple graph of order n, maximum degree ∆ and minimum degree δ ≥ 2. The independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. The girth g(G) is the minimum length of a cycle in G. We establish sharp upper and lower bounds, as functions of n, ∆ and δ, for the independent domination number of graphs G with g(G) ...

For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined the cardinality of minimum set consisting k vertex-disjoint sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has neighbor Vi for all i∈{1,2,…,k}. This invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics Computation 333(15), 2018: 353–361), which aims to compute num...

Let G be a simple graph of order n, maximum degree Δ and minimum degree δ ≥ 2. The independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. The girth g(G) is the minimum length of a cycle in G. We establish best possible upper and lower bounds, as functions of n, Δ and δ, for the independent domination number of graphs G wi...

In this paper, we investigate domination number as well as signed domination numbers of Cay(G : S) for all cyclic group G of order n, where n in {p^m; pq} and S = { a^i : i in B(1; n)}. We also introduce some families of connected regular graphs gamma such that gamma_S(Gamma) in {2,3,4,5 }.

Abstract A set of vertices in a graph is dominating if every vertex or adjacent to . If, addition, an independent set, then set. The domination number the minimum cardinality , while We prove that for all integers it holds connected ‐regular graph, with equality and only result was previously known This affirmatively answers question Babikir Henning.