For a graph ( ) , G V E = , let and denote the set of vertices that are at distance one and two from respectively. A subset 1( ) N v 2 ( ) N v v ( ) D V G ⊆ is said to be a -dominating set of if every vertex 3,2,1 D G v V ∈ satisfies ( ) 3 D w v ≥ where ( ) D w v = 3 { } v D ∩ + 2 ( ) 1 N v D ∩ + ( ) 2 N v D ∩ . The minimum cardinality of a -dominating set of , denoted as 3,2,1 D G ( ) 3,2,1 G ...

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique c...

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 ...

While efficient algorithms for finding minimal distance-k dominating sets exist, finding minimum such sets is NP-hard even for bipartite graphs. This paper presents a distributed algorithm to determine a minimum (connected) distance-k dominating set and a maximum distance-2k independent set of a tree T . It terminates in O(height(T )) rounds and uses O(log k) space. To the best of our knowledge...

For a graph G = (V , E), a subset D V ðGÞ is said to be distance two-dominating set in G if for each vertex u 2 V D, there exists a vertex v 2 D such that dðu; vÞ 2. The minimum cardinality of a distance two-dominating set in G is called a distance twodomination number and is denoted by c2ðGÞ. In this note we obtain various upper bounds for c2ðGÞ and characterize the classes of graphs attaining...

BackgroundGame Theory Interpretation MethodsRandomizationFunctional Lagrange Multipliers ConclusionsReferences

this paper examines the problematization of cultural issues related tothe dual spacization of life. the emergence of the second space of lifeover the first place of life combining local and translocal, national andtransnational capacities and challenges. we discuss the capacities andopportunities occurring through the instant communication industrywhich made the spontaneous connection between a...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...