Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V − S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). Fink and Jaco...

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 ...

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

The independent domination number of a graph G, denoted i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. The graph G is called i-excellent if every vertex of G belongs to some i(G)-set. We provide a constructive characterization of i-excellent trees. c © 2002 Elsevier Science B.V. All rights reserved.

Two new domination parameters for a connected graph G: the weakly convex domination number of G and the convex domination number of G are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

In graph theory, the theory of domination has several applications in various fields science and technology, which is considered as a turn up field research. real life, it extremely important like network desigh, wireless sensor networks,logistics, mobile computing, telecommunication others, Problems with facility location, communication or electrical monitoring can lead to dominance. Undirecte...

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...

The concept of induced paired domination number of a graph was introduced by D.S.Studer, T.W. Haynes and L.M. Lawson11, with the following application in mind. In the guard application an induced paired dominating set represents a configuration of security guards in which each guard is assigned one other as a designated backup with in (as in a paired dominating set), but to avoid conflicts (suc...

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...