On Roman, Global and Restrained Domination in Graphs

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 number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed. © 2010 Springer

Integer Programming Formulations and Probabilistic Bounds for Some Domination Parameters

In this paper, we further study the concepts of hop domination and 2-step domination and introduce the concepts of restrained hop domination, total restrained hop domination, 2-step restrained domination, and total 2-step restrained domination in graphs. We then construct integer programming formulations and present probabilistic upper bounds for these domination parameters

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Fair Secure Dominating Set in the Corona of Graphs

In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results

HIMPUNAN DOMINASI TERKENDALI GRAF HASIL OPERASI AMALGAMASI TITIK DAN SISI PADA GRAF SIKLUS BERORDE SAMA

Teori graf merupakan salah satu ilmu terapan matematika yang hingga kini terus dikembangkan. Salah satu konsep yang dipelajari dalam teori graf adalah himpunan dominasi terkendali yang bertujuan untuk menentukan bilangan dominasi terkendali graf hasil operasi amalgamasi titik dan sisi pada graf siklus berorde sama, atau (a(,,)) dan (a(,,,)). Penelitian ini menggunakan metode pembuktian dengan memanfaatkan teorema yang telah ada, yaitu teorema bilangan dominasi terkendali pada graf siklus dan teorema batas atas dan batas bawah bilangan dominasi terkendali. Adapun hasil yang diperoleh, yaitu:  untuk, dan untuk dimana , dan untukdimana

Bharath hub number of graphs

The mathematical model of a real world problem is designed as Bharath hub number of graphs. In this paper, we study the graph theoretic properties of this variant. Also, we give results for Bharath hub number of join and corona of two connected graphs, cartesian product and lexicographic product of some standard graphs.Publisher's Versio