34 research outputs found
Ramsey-type results on parameters related to domination
There is a philosophy to discover Ramsey-type theorem: given a graph
parameter , characterize the family \HH of graphs which satisfies that
every \HH-free graph has bounded parameter . The classical Ramsey's
theorem deals the parameter as the number of vertices. It also has a
corresponding connected version. This Ramsey-type problem on domination number
has been solved by Furuya. We will use this result to handle more parameters
related to domination.Comment: 12 pages, 1 figures
On the Domination Chain of m by n Chess Graphs
A survey of the six domination chain parameters for both square and rectangular chess boards are discussed
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
Strong Domination Index in Fuzzy Graphs
Topological indices play a vital role in the area of graph theory and fuzzy
graph (FG) theory. It has wide applications in the areas such as chemical graph
theory, mathematical chemistry, etc. Topological indices produce a numerical
parameter associated with a graph. Numerous topological indices are studied due
to its applications in various fields. In this article a novel idea of
domination index in a FG is defined using weight of strong edges. The strong
domination degree (SDD) of a vertex u is defined using the weight of minimal
strong dominating set (MSDS) containing u. Idea of upper strong domination
number, strong irredundance number, strong upper irredundance number, strong
independent domination number, and strong independence number are explained and
illustrated subsequently. Strong domination index (SDI) of a FG is defined
using the SDD of each vertex. The concept is applied on various FGs like
complete FG, complete bipartite and r-partite FG, fuzzy tree, fuzzy cycle and
fuzzy stars. Inequalities involving the SDD and SDI are obtained. The union and
join of FG is also considered in the study. Applications for SDD of a vertex is
provided in later sections. An algorithm to obtain a MSDS containing a
particular vertex is also discussed in the article
Vertex-Edge and Edge-Vertex Parameters in Graphs
The majority of graph theory research on parameters involved with domination, independence, and irredundance has focused on either sets of vertices or sets of edges; for example, sets of vertices that dominate all other vertices or sets of edges that dominate all other edges. There has been very little research on ``mixing\u27\u27 vertices and edges. We investigate several new and several little-studied parameters, including vertex-edge domination, vertex-edge irredundance, vertex-edge independence, edge-vertex domination, edge-vertex irredundance, and edge-vertex independence
Parameters related to fractional domination in graphs.
Thesis (M.Sc.)-University of Natal, 1995.The use of characteristic functions to represent well-known sets in graph theory such as dominating, irredundant, independent, covering and packing sets - leads naturally to fractional versions of these sets and corresponding fractional parameters. Let S be a dominating set of a graph G and f : V(G)~{0,1} the characteristic function of that set. By first translating the restrictions which define a dominating set from a set-based to a function-based form, and then allowing the function f to map the vertex set to the unit closed interval, we obtain the fractional generalisation of the dominating set S. In chapter 1, known domination-related parameters and their fractional generalisations are introduced, relations between them are investigated, and Gallai type results are derived. Particular attention is given to graphs with symmetry and to products of graphs. If instead of replacing the function f : V(G)~{0,1} with a function which maps the vertex set to the unit closed interval we introduce a function f' which maps the vertex set to {0, 1, ... ,k} (where k is some fixed, non-negative integer) and a corresponding change in the restrictions on the dominating set, we obtain a k-dominating function. In chapter 2 corresponding k-parameters are considered and are related to the classical and fractional parameters. The calculations of some well known fractional parameters are expressed as optimization problems involving the k- parameters. An e = 1 function is a function f : V(G)~[0,1] which obeys the restrictions that (i) every non-isolated vertex u is adjacent to some vertex v such that f(u)+f(v) = 1, and every isolated vertex w has f(w) = 1. In chapter 3 a theory of e = 1 functions and parameters is developed. Relationships are traced between e = 1 parameters and those previously introduced, some Gallai type results are derived for the e = 1
parameters, and e = 1 parameters are determined for several classes of graphs. The e = 1 theory is applied to derive new results about classical and fractional domination parameters
Fuzzy Graphs with Equal Fuzzy Domination and Independent Domination Numbers
The basic definitions of fuzzy independent set, fuzzy dominating set,and fuzzy independent dominating sets are discussed. The aim of this paper is to find on what conditions the fuzzy graph has equal domination number and independent domination number. It is discussed briefly and also when the fuzzy graph is domination perfect is proved. Finally, the independent domination number for a connected fuzzy graph is obtained
α-Domination
AbstractLet G=(V,E) be any graph with n vertices, m edges and no isolated vertices. For some α with 0<α⩽1 and a set S⊆V, we say that S is α-dominating if for all v∈V−S,|N(v)∩S|⩾α|N(v)|. The size of a smallest such S is called the α-domination number and is denoted by γα(G). In this paper, we introduce α-domination, discuss bounds for γ1/2(G) for the King's graph, and give bounds for γα(G) for a general α, 0<α⩽1. Furthermore, we show that the problem of deciding whether γα(G)⩽k is NP-complete
Fuzzy Graphs With Equal Fuzzy Domination And Independent Domination Numbers
The basic definitions of fuzzy independent set, fuzzy dominating set,and fuzzy independent dominating sets are discussed. The aim of this paper is to find on what conditions the fuzzy graph has equal domination number and independent domination number. It is discussed briefly and also when the fuzzy graph is domination perfect is proved. Finally, the independent domination number for a connected fuzzy graph is obtained