This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorithms run in time O(n) or O(n log logn) where n is the number of vertices. The results are then extended to s...

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...

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...

BackgroundGame Theory Interpretation MethodsRandomizationFunctional Lagrange Multipliers ConclusionsReferences

The domination number of a graph G = (V,E) is the minimum size of a dominating set U ⊆ V , which satisfies that every vertex in V \ U is adjacent to at least one vertex in U . The notion of a problem kernel refers to a polynomial time algorithm that achieves some provable reduction of the input size. Given a graph G whose domination number is k, the objective is to design a polynomial time algo...

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 ...

Non-dominated sorting, dividing the population into several non-domination levels, is a basic step for many Pareto-based evolutionary multiobjective optimization (EMO) algorithms. Different from the generational scheme, where the selection of next parents is conducted after the generation of a population of offspring, the steady-state scheme updates the parent population whenever a new candidat...