In graph theory, an edge dominating set for a graph g v,e is a subset s. Edge dominating set, secure edge dominating set, secure edge domination number. 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. E is called a dominatingsetif every vertex v2v is either an element of s or is adjacent to an element of s. A dominating set s is a minimaldominatingsetif no proper subset s0. Let d be the minimum dominating set of intuitionistic fuzzy graph g. Two vertices are adjacent if the corresponding subregions share a border. We investigate edge domination number of splitting graph of some standard graphs. 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. 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. Fig 1 the minimal dominating graph of g is an intersection graph on the minimal dominating sets of vertices of g. We now define the edge domination number of jump graph. Independent domination in some wheel related graphs.
Eis an edge dominating set of g, if every edge in e. Gofg is the minimum cardinality taken over all edge dominating sets of g. The set of all minimal dominating sets of a graph gis denoted by mdsg. Make a graph where the vertices are the countries, statesprovinces, counties, or whatever your subregion of interest is. Finding a matching in a bipartite graph can be treated as a network flow problem definition. A dominating set of a graph g is a set of vertices d such that for every vertex v of g, either v. Dominating sets set, by definition, cannot have duplicate elements, else it is not a set. Edge dominating sets in graphs siam journal on applied.
A dominating set of a graph g is a set s of vertices of g such that. It may also be an entire graph consisting of edges without common vertices. 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. The domination number is the number of vertices in a smallest dominating set for g. 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. A study of line graph theory towards line set domination. Cockayne edge domination endvertex endvertices equals the minimum example exists a vertex graph g graph theory grid graphs hence. A set s of vertices of g is a dominating set of g if every vertex in v g. A greedy algorithm for finding a minimum connected dominating set is discussed. An independent set of edges is a set of edges, no two of which have a vertex in common. An edge dominating set is called a minimal edge dominating set or meds if no proper subset of is an edge dominating set. Edge domination in some path and cycle related graphs. Applications of distance 2 dominating sets of graph in.
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. 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. We prove that the edge dominating set problem for graphs is npcomplete even when restricted to planar or bipartite graphs of maximum degree 3. Jul 12, 2006 2015 decision and approximation complexity for identifying codes and locating dominating sets in restricted graph classes. Pdf the edge dominating graph e d g of a graph gv,e is a. Also edge domination number of this graph is obtained. 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.
It has grown rapidly in recent times with a lot of research activities. An independent dominating set in a graph is a set that is both dominating and in dependent. Introduction graph theory is an important branch of mathematics. In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. 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 minimum cardinality of a vedominating set is named vedomination number denoted by. About the edge domination number of the graphs research india. 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. In graph theory, a split of an undirected graph is a cut whose cut set forms a complete bipartite graph.
Motivated by the inverse domination number, there are studies which deals about two. Basavarajappa abstract for any graph g, the line graph l g h is the intersection graph. In this paper, we survey selected results on independent domination in. Given a graph g v, e find a minimum weight dominating set. Siam journal on applied mathematics volume 38, issue 3 10. Functions for finding node and edge dominating sets. An edge dominating set for a graph g is a set d of edges such that each edge of.
It determines how we can calculate the cover or edge covering with the help of an example. Connected edge geodetic domination number of a graph. We assume that the graph g under consideration is nonempty and has no isolated vertices1. 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. Maximum independent set minimum dominating set graph subgraph isomorphism. 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. G of g is the size of the smallest dominating set in g. E such that every edge not in d is adjacent to at least one edge in 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. The edge domination number 0g of g is the minimum cardinality of an edge dominating set in g. Mathematical study of domination in graphs began around 1960. In this paper, we initiate a study of the secure edge domination number and establish some results on this new parameter. 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. 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.
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. Applications of distance 2 dominating sets of graph in networks. Edge domination number on graph variations international. Theory and applications of models of computation, 360372. Every secure edge dominating set of g is an edge dominating set. Equivalently, an independent dominating set is a maximal independent set. A dominating set d is called independent dominating set of lg, if d is also. 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. The size of a minimum dominating set in a graph g is called the domination number of g and is denoted by. 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. In graph theory, an edge dominating set for a graph gv,e is a subset s.
The edge domination number of connected graphs araya chaemchan. 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. Independent dominating sets have been studied extensively in the literature. G is the number of vertices in a smallest dominating set for g. Mynhardt chapter chordal graphs classes of graphs closed neighborhood complete graph comput connected dominating set connected graph. Graph colouring is just one of thousands of intractable.
The edge domination number of is the minimum cardinality taken over all edge dominating sets of. The notes form the base text for the course mat62756 graph theory. E such that every edge not in s is adjacent to at least one edge in s. That is, for every vertex u 2v s, there exists a vertex v 2s such that uv 2e. In graph theory, the problem concerning domination of graphs or networks. Given a graph g v,e, a matching m in g is a set of pairwise nonadjacent edges, none of which are loops. G is the minimum cardinality of a dominating set in g, and a dominating set s of minimum cardinality is called a. A minimum dominating set in a graph g is a dominating. 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. In this paper, we initiate some definitions onedge dominating set concerning intuitionistic fuzzy sets. An edge dominating set is also known as a line dominating set. In graph theory, an edge dominating set for a graph g v, e is a subset d. 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. The independent domination number and the notation were introduced by.
V is a vedominating set if every edges of a graph g is vedominated by at least one vertex of s 2, 5. The following is a brief history of domination in graphs. Pdf applications of dominating set of graph in computer. A set of vertices in a graph is called an independent dominating set if is both an independent set and a dominating set of. Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. The dominating set problem concerns testing whether. The independent domination number of g, denoted by ig, is the minimum cardinality of. Thus the vertices of lg are the edges of g, with two vertices of adjacent whenever the corresponding edges of g are. An independent edge dominating set is an independent set of edges which is. Media in category dominating set graph theory the following 12 files are in this category, out of 12 total. 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.
For a graph, a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in. 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. Gofg is the minimum cardinality of an edge dominating set of 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 connected edge domination number of g is the minimum cardinality of a connected edge dominating set. The independent domination number and the notation were introduced by cockayne and hedetniemi 1974, 1977. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles. Research article edge domination in some path and cycle. 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. 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. In this paper, we survey selected results on independent domination in graphs. 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. About the edge domination number of the graphs 97 1 is already proved.
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. 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. 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. This concept was introduced by kulli and janakiram 4. The domination number gammag is the number of vertices in a smallest dominating set for g. 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. V is a dominating set if nsv, or equivalently, every vertex in v. In the last few decades, at the international level, one third of the mathematics research papers are from graph theory and combinatorics. An edge e evdominates a vertex v which is a vertex of e, as well as every vertex adjacent to v 2. In the last few decades, at the international level, one third of the mathematics research papers are from graph theory. Figures ad are examples of edge dominating sets thick red lines. Index terms edge domination number, splitting graph, edge splitting graph.
57 344 579 844 1397 964 478 1226 360 956 74 327 750 1000 1262 115 157 158 869 164 963 741 265 1025 1195 1004 452 227 539 1018 1643 237 231 1460 1578 1405 593 105 42 811 627 577 517 803 478 868