On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs

Domination problems in directed graphs and inducibility of nets

In this thesis we discuss two topics: domination parameters and inducibility. In the first chapter, we introduce basic concepts, definitions, and a brief history for both types of problems. We will first inspect domination parameters in graphs, particularly independent domination in regular graphs and we answer a question of Goddard and Henning. Additionally, we provide some constructions for graphs regular graphs of small degree to provide lower bounds on the independent domination ratio of these classes of graphs. In Chapter 3 we expand our exploration of independent domination into the realm of directed graphs. We will prove several results including providing a fastest known algorithm for determining existence of an independent dominating set in directed graphs with minimum in degree at least one and period not eqeual to one. We also construct a set of counterexamples to the analogue of Vizing\u27s Conjecture for this setting. In the fourth chapter, we pivot from independent domination to split domination in directed graphs, where we introduce the split domination sequence. We will determine that almost all possible split domination sequences are realizable by some graphs, and state several open questions that would be of interest to continue on this field. In the fifth chapter we will provide a brief introduction to Flag Algebras, then determine the unique maximizer of induced net graphs in graphs of certain orders

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Graphs with equal domination and independent domination number

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exact values of their domination and independent domination numbers are determined.Publisher's Versio

Independent Domination in Some Wheel Related Graphs

A set S of vertices in a graph G is called an independent dominating set if S is both independent and dominating. The independent domination number of G is the minimum cardinality of an independent dominating set in G . In this paper, we investigate the exact value of independent domination number for some wheel related graphs

Independent Domination Of Subcubic Graphs

Let G be a simple graph. The independent domination number i(G) is the minimum cardinality among all maximal independent sets of G. A graph is subcubic whenever the maximum degree is at most three. In this paper, we will show that the independent domination number of a connected subcubic graph of order n having minimum degree at least two is at most 3(n+1)/7, providing a sharp upper bound for subcubic connected graphs with minimum degree at least two

On resource placements and fault-tolerant broadcasting in toroidal networks

Parallel computers are classified into: Multiprocessors, and multicomputers. A multiprocessor system usually has a shared memory through which its processors can communicate. On the other hand, the processors of a multicomputer system communicate by message passing through an interconnection network. A widely used class of interconnection networks is the toroidal networks. Compared to a hypercube, a torus has a larger diameter, but better tradeoffs, such as higher channel bandwidth and lower node degree. Results on resource placements and fault-tolerant broadcasting in toroidal networks are presented. Given a limited number of resources, it is desirable to distribute these resources over the interconnection network so that the distance between a non-resource and a closest resource is minimized. This problem is known as distance-d placement. In such a placement, each non-resource must be within a distance of d or less from at least one resource, where the number of resources used is the least possible. Solutions for distance-d placements in 2D and 3D tori are proposed. These solutions are compared with placements used so far in practice. Simulation experiments show that the proposed solutions are superior to the placements used in practice in terms of reducing average network latency. The complexity of a multicomputer increases the chances of having processor failures. Therefore, designing fault-tolerant communication algorithms is quite necessary for a sufficient utilization of such a system. Broadcasting (single-node one-to-all) in a multicomputer is one of the important communication primitives. A non-redundant fault-tolerant broadcasting algorithm in a faulty toroidal network is designed. The algorithm can adapt up to (2n-2) processor failures. Compared to the optimal algorithm in a fault-free n-dimensional toroidal network, the proposed algorithm requires at most 3 extra communication steps using cut through packet routing, and (n + 1) extra steps using store-and-forward routing