The six basic parameters relating to domination, independence and irredundance satisfy a chain of inequalities given by ir ≤ γ ≤ i ≤ β0 ≤ Γ ≤ IR where ir, IR are the irredundance and upper irredundance numbers, γ,Γ are the domination and upper domination numbers and i, β0 are the independent domination number and independence number respectively. In this paper, we introduce the concept of indep...

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u ∈ V (D) \ S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γd(G), which is the maximum directed domination number γ(D) ove...

‎A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$‎. ‎The total domination number of a graph $G$‎, ‎denoted by $gamma_t(G)$‎, ‎is~the minimum cardinality of a total dominating set of $G$‎. ‎Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International ournal of Graphs and Combinatorics 1 (2004)‎, ‎6...

For any two vertices u and v in a graph G (digraph D, respectively), a u-v geodesic is a shortest path between u and v (from u to v, respectively). Let I(u, v) (ID(u, v), respectively) denote the set of all vertices lying on a u-v geodesic. For a vertex subset S, let IG(S) (ID(S), respectively) denote the union of all IG(u, v) (ID(u, v), respectively) for u, v ∈ S. The geodetic number g(G) (g(D...

let $g$ be a simple graph of order $n$. the domination polynomial of $g$ is the polynomial $d(g, x)=sum_{i=gamma(g)}^{n} d(g,i) x^{i}$, where $d(g,i)$ is the number of dominating sets of $g$ of size $i$ and $gamma(g)$ is the domination number of $g$. in this paper we present some families of graphs whose domination polynomials are unimodal.

Let G = (V , E) be a graph. A subset D ⊆ V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a...

A set S of vertices of a graphG = (V,E) is a dominating set if every vertex of V (G)\S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal ...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...