In this paper we have introduced new concepts such as global vertex edge dominating set, global vertex edge irredundant set, global independent vertex edge dominating set. The independent domination number of g, denoted by ig, is the minimum cardinality of. Theory and applications of models of computation, 360372. All graphs have edge sets, yet the edge set of the multigraph would have to contain duplicate sets in its edge set to properly represent its edges, but then it wouldnt have an edge set, because a set cannot possibly have duplicate elements. We assume that the graph g under consideration is nonempty and has no isolated vertices1. Maximum independent set minimum dominating set graph subgraph isomorphism. Pdf the edge dominating graph e d g of a graph gv,e is a. A dominating set of a graph g is a set s of vertices of g such that every vertex not in s is adjacent to a vertex in s. G is the minimum cardinality of a dominating set in g, and a dominating set s of minimum cardinality is called a. It has grown rapidly in recent times with a lot of research activities.
For a graph, a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in. About the edge domination number of the graphs 97 1 is already proved. G is the number of vertices in a smallest dominating set for g. We investigate edge domination number of splitting graph of some standard graphs. In this paper, we survey selected results on independent domination in. Mynhardt chapter chordal graphs classes of graphs closed neighborhood complete graph comput connected dominating set connected graph. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. In the last few decades, at the international level, one third of the mathematics research papers are from graph theory. A dominating set d is called independent dominating set of lg, if d is also. Pdf applications of dominating set of graph in computer.
An edge dominating set is called a minimal edge dominating set or meds if no proper subset of is an edge dominating set. V is a dominating set if nsv, or equivalently, every vertex in v. A subset x of edges in a graph g is called an edge dominating set of g if every edge not in x is adjacent to some edge in x. In this paper, we survey selected results on independent domination in graphs. Introduction graph theory is an important branch of mathematics.
A dominating set in a graph g is a subset of vertices s v such that each vertex in v is either in s or is adjacent to some vertex in s. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles. That is, for every vertex u 2v s, there exists a vertex v 2s such that uv 2e. Solving the weighted efficient edge domination problem on bipartite. Given a graph g v,e, a matching m in g is a set of pairwise nonadjacent edges, none of which are loops. Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. An independent set of edges is a set of edges, no two of which have a vertex in common. Two vertices are adjacent if the corresponding subregions share a border. The minimum intuitionistic fuzzy cardinality of all edge dominating set of intuitionistic fuzzy graph g is known as edge domination number and it is denoted by. In this paper, we initiate a study of the secure edge domination number and establish some results on this new parameter. Thus the vertices of lg are the edges of g, with two vertices of adjacent whenever the corresponding edges of g are.
The following is a brief history of domination in graphs. In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset. Given a graph g v, e find a minimum weight dominating set. An edge dominating set is also known as a line dominating set. Let d be the minimum dominating set of intuitionistic fuzzy graph g. The domination number gammag is the number of vertices in a smallest dominating set for g. In graph theory, a split of an undirected graph is a cut whose cut set forms a complete bipartite graph. Pdf edge dominating functions of quadratic residue cayley. G of g is the size of the smallest dominating set in g. An independent edge dominating set is an independent set of edges which is. In 5, the concept of common minimal dominating graph of g was defined as the graph having same vertex set as g with two vertices adjacent if there is.
The edge domination number of connected graphs araya chaemchan. The connected edge domination number of g is the minimum cardinality of a connected edge dominating set. A set of vertices in a graph is called an independent dominating set if is both an independent set and a dominating set of. A dominating set s is a minimaldominatingsetif no proper subset s0. The theory of edge dominating functions in quadratic residue cayley graphs helps in finding optimal global and local alignments for the smooth conduction of a work and improves the ability of a. In this paper, we initiate some definitions onedge dominating set concerning intuitionistic fuzzy sets. The edge domination number 0g of g is the minimum cardinality of an edge dominating set in g. An edge dominating set d is a set of edges from the edge set e in graph g where. Connected edge dominating set an edge dominating set f of a graph g is a connected edge dominating set if the induced subgraph is connected. Eis an edge dominating set of g, if every edge in e. Edge domination number on graph variations international. This concept was introduced by kulli and janakiram 4. Mynhardt chapter chordal graphs classes of graphs closed neighborhood complete graph comput connected dominating set connected graph contradiction corollary cycle defined degv denote diamg discrete math domatic dominating function domination in graphs domination number dtg e. E is called a dominatingsetif every vertex v2v is either an element of s or is adjacent to an element of s.
Graph theory objective questions and answers given a directed graph with positive edge weights, find the minimum cost path regarding your first question, i have a nonlinear objective and additional by posting your answer, you agree to. Research article edge domination in some path and cycle. It determines how we can calculate the cover or edge covering with the help of an example. In a graph g, a subset s of v is global vertexedge dominating set if s is vertexedge dominating set in both g and g in this paper we have introduced new concepts such as global vertexedge dominating set, global vertexedge irredundant set, global independent vertexedge dominating set. Motivated by the inverse domination number, there are studies which deals about two. Index terms edge domination number, splitting graph, edge splitting graph. Equivalently, an independent dominating set is a maximal independent set. The splits of a graph can be collected into a treelike structure called the split decomposition or join decomposition, which can be constructed in linear time.
A geodetic domination set of g is a subset of vg which is both geodetic and dominating set of g. Connected edge geodetic domination number of a graph. The notes form the base text for the course mat62756 graph theory. A dominating set of a graph g is a set s of vertices of g such that. Applications of distance 2 dominating sets of graph in networks. Figures ad are examples of edge dominating sets thick red lines. Graph colouring is just one of thousands of intractable. A set s of vertices of g is a dominating set of g if every vertex in v g. Suppose that there is some edge vy which is also in f 1 then f 1xy is an edge dominating set of g, which is a contradiction. The minimum cardinality of a vedominating set is named vedomination number denoted by. Let g v, e be a connected graph, a subset s of eg is called a boundary edge dominating set if every edge of e. Independent domination in some wheel related graphs. The size of a minimum dominating set in a graph g is called the domination number of g and is denoted by. The aim of the paper is to impart the importance of graph theoretical concepts and the applications of domination in graphs to various real life situations in the areas of science and engineering.
Edge dominating set, secure edge dominating set, secure edge domination number. It can also be defined as a set f of edges g is called an edge dominating set of g if for every edge, there exists an edge such that and have a vertex in common. The edge domination number of is the minimum cardinality taken over all edge dominating sets of. Basavarajappa abstract for any graph g, the line graph l g h is the intersection graph. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. An independent dominating set in a graph is a set that is both dominating and independent. The dominating set problem concerns testing whether. Functions for finding node and edge dominating sets. In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. Applications of distance 2 dominating sets of graph in.
Aug 21, 2014 this work is supported by nsfc 11161046, xinjiang young talent project 20721012, and research found of henan normal university qd42. Gofg is the minimum cardinality of an edge dominating set of g. In the last few decades, at the international level, one third of the mathematics research papers are from graph theory and combinatorics. E such that every edge not in s is adjacent to at least one edge in s. In graph theory, an edge dominating set for a graph g v, e is a subset d. Edge domination in graph is a growing area for researcher and mathematician. Also edge domination number of this graph is obtained. About the edge domination number of the graphs research india. Jul 12, 2006 2015 decision and approximation complexity for identifying codes and locating dominating sets in restricted graph classes. A total dominating set of a graph g is a set of the vertex set v of g such that every vertex of g is adjacent to a vertex in s. We prove that the edge dominating set problem for graphs is npcomplete even when restricted to planar or bipartite graphs of maximum degree 3. The independent domination number and the notation were introduced by cockayne and hedetniemi 1974, 1977. In graph theory, an edge dominating set for a graph g v,e is a subset s. E, a subset f of e is called an edge dominating set of g if every edge not in f is adjacent to some edge in f.
A dominating set in a graph g is a subset of vertices of g such that every vertex outside the subset has neighbour in it. V is a vedominating set if every edges of a graph g is vedominated by at least one vertex of s 2, 5. Fig 1 the minimal dominating graph of g is an intersection graph on the minimal dominating sets of vertices of g. An independent dominating set in a graph is a set that is both dominating and in dependent. E is the graph whose vertex set is in onetoone correspondence with the elements of the set e such that two vertices of lg are adjacent if and only if they correspond to two adjacent edge of g.
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges, that is, edges that have the same end nodes. It may also be an entire graph consisting of edges without common vertices. 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. Independent dominating sets have been studied extensively in the literature. An edge e evdominates a vertex v which is a vertex of e, as well as every vertex adjacent to v 2. Media in category dominating set graph theory the following 12 files are in this category, out of 12 total. Making a dominating set of a graph connected hengzhe li college of mathematics and information science. Matching graph theory in the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. The domination number is the number of vertices in a smallest dominating set for g. Cockayne edge domination endvertex endvertices equals the minimum example exists a vertex graph g graph theory grid graphs hence. The set of all minimal dominating sets of a graph gis denoted by mdsg.
Siam journal on applied mathematics volume 38, issue 3 10. Gofg is the minimum cardinality taken over all edge dominating sets of g. In graph theory, an edge dominating set for a graph gv,e is a subset s. Edge domination in some path and cycle related graphs. Mathematical study of domination in graphs began around 1960. Every secure edge dominating set of g is an edge dominating set. Definition a set f of edges in a graph is called an edge dominating set of g if every edge in is adjacent to at least one edge in f. A study of line graph theory towards line set domination.
Finding a matching in a bipartite graph can be treated as a network flow problem definition. A minimum dominating set in a graph g is a dominating. Transferring this result into the line graph we have that the minimum dominating set problem is npcomplete for 2k 1regular graphs, k 3, that is regular graphs with even degree at least four. E such that every edge not in d is adjacent to at least one edge in d. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. A subset d of e is said to be an edge dominating set eds of g if every edge in e d is adjacent to some edge in d. Graph colouring graph theory in the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. A greedy algorithm for finding a minimum connected dominating set is discussed. Make a graph where the vertices are the countries, statesprovinces, counties, or whatever your subregion of interest is. Dominating sets set, by definition, cannot have duplicate elements, else it is not a set. In graph theory, a dominating set for a graph g v, e is a subset d of v such that every vertex not in d is adjacent to at least one member of d. The smallest number of colours needed to colour the map is the chromatic number of the graph this is probably why we refer to it as a colouring.
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