We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

In the Domination Game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set A∗ dominates the whole graph. The Dominator plays to minimise the size of the set A∗ while the Staller plays to maximise it. A graph is D-trivial if when the Dominator plays first and both p...

The major factors of insecurity in the East, away from the security of geopolitical area were because of the formation of political management in the nineteenth century colonial style. The security factor in the vast region of the East that encompasses at least three countries of Iran and Afghanistan and Pakistan is very important and highly effective. It reached its peak colonial Anglo-Russian...

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...

Matching Pennies game (MPG) and and Rock-Paper-Scissors game (RPS) are the two elementary games. These two games are used to illustrate the concept (mixed strategy Nash equilibrium) since its very begin of the modern game theory. Under the traditional multi-person random pairwise-matching experimental protocol, using the definitive measurement, we have found the processes not in equilibrium but...

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...

In this paper, we introduce the closed domination in graphs. Some interesting relationships are known between domination and closed domination and between closed domination and the independent domination. It is also shown that any triple m, k and n of positive integers with 3 ≤ m ≤ k ≤ n are realizable as the domination number, closed domination number and independent domination number, respect...