1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are ob...

A k-dominating set of a graph G is a set S of vertices of G such that every vertex outside of S has k neighbors in S. The k-domination number of G, written γk(G), is the size of the smallest k-dominating set in G. In this paper, we derive sharp upper and lower bounds on γk(G) + γk(G) and γk(G)γk(G), where G is the complement of G. We use the results for k = 2 to prove a conjecture of Alon, Balo...

a {em roman dominating function} on a graph $g$ is a function$f:v(g)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}a {em restrained roman dominating}function} $f$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

In a graph G, a vertex is said to dominate itself and all vertices adjacent to it. For a positive integer k, the k-tuple domination number γ×k(G) of G is the minimum size of a subset D of V (G) such that every vertex in G is dominated by at least k vertices in D. To generalize/improve known upper bounds for the k-tuple domination number, this paper establishes that for any positive integer k an...

This thesis deals with the following three independent problems. Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgra...

In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e....

Let γ(n, δ) denote the largest possible domination number for a graph of order n and minimum degree δ. This paper is concerned with the behavior of the right side of the sequence n = γ(n, 0) ≥ γ(n, 1) ≥ · · · ≥ γ(n, n− 1) = 1. We set δk(n) = max{δ | γ(n, δ) ≥ k}, k ≥ 1. Our main result is that for any fixed k ≥ 2 there is a constant ck such that for sufficiently large n, n− ckn ≤ δk+1(n) ≤ n− n...

‎let $g=(v(g),e(g))$ be a graph‎, ‎$gamma_t(g)$. let $ooir(g)$ be the total domination and oo-irredundance number of $g$‎, ‎respectively‎. ‎a total dominating set $s$ of $g$ is called a $textit{total perfect code}$ if every vertex in $v(g)$ is adjacent to exactly one vertex of $s$‎. ‎in this paper‎, ‎we show that if $g$ has a total perfect code‎, ‎then $gamma_t(g)=ooir(g)$‎. ‎as a consequence, ...