E of minimum cardinality is sought in a given graph g v,e with a demand vector b. The edge domination number 0g of g is the minimum cardinality of an edge dominating set in g. E such that every edge not in s is adjacent to at least one edge in s. 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. Edge domination in some path and cycle related graphs open.
Kulli, entire edge dominating transformation graphs, international journal of. In this sense, for connected graphs, any minimal vertex cover is a dominating set, and therefore always contains a minimal dominating set. Jul 12, 2006 2015 decision and approximation complexity for identifying codes and locating dominating sets in restricted graph classes. In this paper we prove that the perfect domination number of book graph and stacked. A singlevertex dominating set may also be called a universal vertex. In graph theory, an edge dominating set for a graph g v,e is a subset s. Many different types of domination have been researched extensively this paper explores applications of. A vertex cover may not be a dominating set if you have a degree zero vertex outside your vertex cover. Graph theory 121 circuit a circuit is a path that begins and ends at the same vertex. The middle edge dominating graph m ed g of a graph gv, e is a graph with the vertex set e.
We note, for later reference, that every minimal dominating set is a. A set, by definition, cannot have duplicate elements, else it is not a set. Part of the lecture notes in computer science book series lncs, volume 8973. An independent edge dominating set is in fact a maximal matching and a minimum independent edge dominating set is a minimum maximal matching discussed in 4. The minimum fuzzy cardinality of an edge dominating set is g is called the edge domination number of g and it is. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. V is a dominating set of g if every vertex is either in s or joined by an edge to some vertex in s. Gofg is the minimum cardinality of an edge dominating set of g. Each edge connects a vertex to another vertex in the graph or itself, in the case of a loopsee answer to what is a loop in graph theory.
We investigate edge domination number of splitting graph of some standard graphs. In this paper, we initiate some definitions onedge dominating set concerning intuitionistic fuzzy sets. Introduction graph theory is an important branch of mathematics. Domination numbers of graphs containing vertexdisjoint. Two vertices are adjacent if the corresponding subregions share a border. A survey on the effect of graph operations on the domination. For a graph, a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in. Edge weighting functions on semitotal dominating sets, graphs. Edge domination in some path and cycle related graphs hindawi. A graph with n nodes and n1 edges that is connected. Part of the lecture notes in computer science book series lncs, volume 7676. A set f of the edges of a graph g is an edge dominating set if each edge of e\f is adjacent to an edge of f. In a graph, a domination set is a subset s of the vertices such that every vertices of v s is adjacent to a vertex of s.
The domination number of a graph is the number of vertices in the smallest dominating set. Inverse split edge domination in fuzzy graphs definition 3. Let d be the minimum dominating set of intuitionistic fuzzy graph g. Perfect domination in book graph and stacked book graph. A subset s of x is said to be an edge domination set in g if for every edge in xs is adjacent to atleast one effective edge in s. A greedy algorithm for finding a minimum connected dominating set is discussed. An edge dominating set of a graph g v, e is a subset m. A geodetic domination set of g is a subset of vg which is both geodetic and dominating set of g.
S where s is the set of all minimal edge dominating set g and with two vertices u,v. E such that every edge not in d is adjacent to at least one edge in d. A set d is minimal dominating set of a graph if for any vertex v in d, the set d v is not a dominating set of the graph, hence not every minimal dominating set is minimum but every minimum. Connected edge geodetic domination number of a graph. 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. 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. An independent dominating set in a graph is a set that is both dominating and in dependent. Efficient dominating and edge dominating sets for graphs and hypergraphs. Applications of distance 2 dominating sets of graph in networks. A graph with maximal number of edges without a cycle. Recently many domination parameters are given in the books by kulli in 2,3, 4. V is a vedominating set if every edges of a graph g are vedominated. This paper considers a variant of eds with extensions of multiple and connected dominations combined. A circuit starting and ending at vertex a is shown below.
A dominating set of a graph g is a set s of vertices of g such that. The edge dominating set problem eds is to compute a minimum edge set such that every edge is dominated by some edge in it. A simple and fast algorithm for maximum independent set in 3degree graphs. Then there exists such that e and f are adjacent and is an edge dominating set of g. An improved algorithm for parameterized edge dominating set. It has grown rapidly in recent times with a lot of research activities.
Algorithm atleast atmost automorphism bipartite graph called clique complete graph connected graph contradiction corresponding cut vertex cycle darithmetic definition degree sequence deleting denoted digraph displayed in figure divisor graph dominating set edge of g end vertex euler tour eulerian example exists frontier edge g contains g is. Gofg is the minimum cardinality taken over all edge dominating sets of g. Mynhardt chapter chordal graphs classes of graphs closed neighborhood complete graph comput connected dominating set connected graph. An independent set of edges is a set of edges, no two of which have a vertex in common.
In the last few decades, at the international level, one third of the mathematics research papers are from graph theory and combinatorics. Perfect edge domination and efficient edge domination in graphs. An independent line set is said to be the maximal independent line set of a graph g if no other edge of g can be added to l. Siam journal on applied mathematics volume 38, issue 3 10. Domination in graphs has been an extensively researched branch of graph theory. G is the number of vertices in a smallest dominating set for g.
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. A dominating set may not be a vertex cover if there is an edge, say e u,v, where u and v are both outside the dominating set. If f is a secure total edge dominating set of a graph g, then f is a secure edge dominating set of g. Independent dominating sets have been studied extensively in the literature. A graph in which any two nodes are connected by a unique path path edges may only be traversed once. By a graph g v, e we mean a finite, undirected graph with neither loops nor multiple edges. Equivalently, an independent dominating set is a maximal independent set. Entire edge dominating transformation graphs ijarcst. In graph theory, an edge dominating set for a graph g v, e is a subset d. Edge dominating sets in graphs siam journal on applied. As pointed out in 8, the size of the minimum edge dominating set of a graph g is. 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.
A graph with no cycle in which adding any edge creates a cycle. The size of a minimum dominating set in a graph g is called the domination number of g and is denoted by. V is a dominating set if nsv, or equivalently, every vertex in v. The domination number is the number of vertices in a smallest dominating set for g. 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. But domination number was studied in the name of coefficient of external stability. The edge domination number of is the minimum cardinality taken. Media in category dominating set graph theory the following 12 files are in this category, out of 12 total. Pdf edge domination in some path and cycle related graphs.
An independent edge dominating set is an edge dominating set in which no two elements are adjacent. Graph colouring book, and i will keep it for a future reference. 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. A subset d of e is called a perfect edge dominating set if every edge of e. I would think that your example is more about dominating sets and vertex covers of minimum size. V is a semipaired dominating set of g if every vertex in v \ s is. Index terms edge domination number, splitting graph, edge splitting graph. Connected a graph is connected if there is a path from any vertex to any other vertex.
Cockayne edge domination endvertex endvertices equals the minimum example exists a vertex graph g graph theory grid graphs hence. The study of domination number in graph theory was introduced by claude berge in 1958, on his book theory of graphs and its applications 1. Graphs consist of a set of vertices v and a set of edges e. Figures ad are examples of edge dominating sets thick red lines. Dominating sets graph with a minimal number of edges which is connected. An independent dominating set in a graph is a set that is both dominating and independent. The paper concentrates on the domination in graphs with application in a graph g v, e, s. G is the minimum cardinality of a dominating set in g, and a dominating set s of minimum cardinality is called a. Suppose f is a secure total edge dominating set of. Applications of distance 2 dominating sets of graph in. Efficient dominating and edge dominating sets for graphs and.
The edge domination number of is the minimum cardinality taken over all edge dominating sets of. 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. In the last few decades, at the international level, one third of the mathematics. Jul 29, 2015 the middle edge dominating graph m ed g of a graph gv, e is a graph with the vertex set e. Edge domination in some path and cycle related graphs. Part of the lecture notes in computer science book series lncs, volume 7464. The dominating set problem concerns testing whether. We prove that the edge dominating set problem for graphs is npcomplete even when restricted to planar or bipartite graphs of maximum degree 3. We consider a multiple domination version of the edge dominating set problem, called the beds problem, where an edge set d. At the first time the name dominating set and domination number was used by oyestein ore in 2.
Make a graph where the vertices are the countries, statesprovinces, counties, or whatever your subregion of interest is. Many different types of domination have been researched extensively this paper explores applications of dominating sets. However l 1 is not an independent line set, as for making an independent line set, there should be at least two edges. The main object of this article is to study the domination numbers of graph containing vertex disjoint cycles with some identities of domination set and independent set. Pdf for a graph g v, e, a subset f of e is called an edge dominating set of g if. An independent edge dominating set is an independent. On approximating connected 2edge dominating set by a tree.
Domination in graphs usf scholar commons university of south. An edge dominating set is also known as a line dominating set. In this paper, we survey selected results on independent domination in graphs. The vertex cover covers all the edges, but the degree zero vertex is not adjacent to the vertex cover. Theory and applications of models of computation, 360372. 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. An independent line set is said to be the maximal independent line set of a graph g if.
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