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Edge-Vertex Dominating Sets and Edge-Vertex Domination Polynomials of Cycles  [PDF]
A. Vijayan, J. Sherin Beula
Open Journal of Discrete Mathematics (OJDM) , 2015, DOI: 10.4236/ojdm.2015.54007
Abstract: Let G = (V, E) be a simple graph. A set S \"\"E(G) is an edge-vertex dominating set of G (or simply an ev-dominating set), if for all vertices v \"\"V(G); there exists an edge e\"\"S such that e dominates v. Let \"\" denote the family of all ev-dominating sets of \"\" with cardinality i. Let \"\". In this paper, we obtain a recursive formula for \"\". Using this recursive formula, we construct the polynomial,\"\" , which we call edge-vertex domination polynomial of \"\" (or simply an ev-domination polynomial of \"\") and obtain some properties of this polynomial.
Edge Domination in Some Path and Cycle Related Graphs  [PDF]
S. K. Vaidya,R. M. Pandit
ISRN Discrete Mathematics , 2014, DOI: 10.1155/2014/975812
Abstract: For a graph , a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in . The edge domination number of is the minimum cardinality taken over all edge dominating sets of . Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles. 1. Introduction The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because of its many and varied applications in such fields as linear algebra and optimization, design and analysis of communication networks, and social sciences and military surveillance. Many variants of dominating models are available in the existing literature. For a comprehensive bibliography of papers on the concept of domination, readers are referred to Hedetniemi and Laskar [1]. The present paper is focused on edge domination in graphs. We begin with simple, finite, connected, and undirected graph of order . The set of vertices in a graph is called a dominating set if every vertex is either an element of or is adjacent to an element of . A dominating set is a minimal dominating set (or MDS) if no proper subset is a dominating set. The minimum cardinality of a dominating set of is called the domination number of which is denoted by , and the corresponding dominating set is called a -set of . The open neighborhood of is the set of vertices adjacent to , and the set is the closed neighborhood of . For any real number , denotes the smallest integer not less than and denotes the greatest integer not greater than . An edge of a graph is said to be incident with the vertex if is an end vertex of . In this case, we also say that is incident with . Two edges and which are incident with a common vertex are said to be adjacent. In a graph , a vertex of degree one is called a pendant vertex, and an edge incident with a pendant vertex is called a pendant edge. A subset is an edge dominating set if each edge in is either in or is adjacent to an edge in . An edge dominating set is called a minimal edge dominating set (or MEDS) if no proper subset of is an edge dominating set. The edge domination number is the minimum cardinality among all minimal edge dominating sets. The concept of edge domination was introduced by Mitchell and Hedetniemi [2] and it is explored by many researchers. Arumugam and Velammal [3] have discussed the edge domination in graphs while the fractional edge domination in graphs is discussed in Arumugam and Jerry [4]. The complementary edge domination in graphs is studied by Kulli and
On the Domination Polynomial of Some Graph Operations  [PDF]
Saeid Alikhani
ISRN Combinatorics , 2013, DOI: 10.1155/2013/146595
Abstract: Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of is called the domination root of . In this paper, we study the domination polynomial of some graph operations. 1. Introduction Let be a simple graph. For any vertex , the open neighborhood of is the set and the closed neighborhood is the set . For a set , the open neighborhood of is and the closed neighborhood of is . A set is a dominating set if , or equivalently, every vertex in is adjacent to at least one vertex in . An -subset of is a subset of of cardinality . Let be the family of dominating sets of which are -subsets and let . The polynomial is defined as domination polynomial of [1]. This polynomial has been introduced by the author in his Ph.D. thesis in 2009 [2]. A root of is called a domination root of . More recently, domination polynomial has found application in network reliability [3]. For more information and motivation of domination polynomial and domination roots refer to [1, 2]. The join of two graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and . The corona of two graphs and , is the graph formed from one copy of and copies of , where the th vertex of is adjacent to every vertex in the th copy of [4]. the Cartesian product of two graphs and is denoted by , is the graph with vertex set and edges between two vertices and if and only if either and or and . In this paper, we study the domination polynomials of some graph operations. 2. Main Results As is the case with other graph polynomials, such as chromatic polynomials and independence polynomials, it is natural to consider the domination polynomial of composition of two graphs. It is not hard to see that the formula for domination polynomial of join of two graphs is obtained as follows. Theorem 1 (see [1]). Let and be graphs of orders and , respectively. Then It is obvious that this operation of graphs is commutative. Using this product, one is able to construct a connected graph with the number of dominating sets , where is an arbitrary odd natural number; see [5]. Let to consider the corona of two graphs. The following theorem gives us the domination polynomial of graphs of the form which is the first result for domination polynomial of specific corona of two graphs. Theorem 2 (see [1]). Let be a graph. Then if and only if for some graph of order . It is easy to see that the corona operation of two graphs does not have the commutative property. The following theorem gives
Some New Perspectives on Global Domination in Graphs  [PDF]
S. K. Vaidya,R. M. Pandit
ISRN Combinatorics , 2013, DOI: 10.1155/2013/201654
Abstract: A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . Here we explore the possibility to relate the domination number of graph and the global domination number of the larger graph obtained from by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results. 1. Introduction The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it. Many variants of dominating sets are available in the existing literature. This paper is focused on global domination in graphs. We begin with simple, finite, and undirected graph with . The set is called a dominating set if . A dominating set is called a minimal dominating set (MDS) if no proper subset of is a dominating set. The minimum cardinality of a dominating set in is called the domination number of denoted by , and the corresponding dominating set is called a -set of . The complement of is the graph with vertex set in which two vertices are adjacent in if and only if they are not neighbors in . A dominating set of is called a global dominating set if it is also a dominating set of . The global domination number is the minimum cardinality of a global dominating set of . The concept of global domination in a graph was introduced by Sampathkumar [1]. This concept is remained in focus of many researchers. For example, the global domination number of boolean function graph is discussed by Janakiraman et al. [2]. The NP completeness of global domination problems is discussed by Carrington [3] and by Carrington and Brigham [4]. The global domination number for the larger graphs obtained from the given graph is discussed by Vaidya and Pandit [5] while Kulli and Janakiram [6] have introduced the concept of total global dominating sets. The discussion on global domination in graphs of small diameters is carried out by Gangadharappa and Desai [7]. The wheel is defined to be the join where . The vertex corresponding to is known as apex vertex, and the vertices corresponding to cycle are known as rim vertices. Duplication of an edge of a graph produces a new graph by adding an edge such that and . The shadow graph of a connected graph is constructed by taking two copies of , say and . Join each vertex in to the neighbors of the corresponding vertex in . A vertex switching of a graph is the graph obtained by taking a vertex of , removing all
Some New Results on Distance -Domination in Graphs  [PDF]
Samir K. Vaidya,Nirang J. Kothari
International Journal of Combinatorics , 2013, DOI: 10.1155/2013/795401
Abstract: We determine the distance -domination number for the total graph, shadow graph, and middle graph of path . 1. Introduction We begin with finite, connected, and undirected graphs, without loops or multiple edges. A dominating set of a graph is a set of vertices of such that every vertex of is adjacent to some vertex of . The domination number is the minimum cardinality of a dominating set of . Further, the open neighbourhood of is the set . The closed neighbourhood of is the set . The distance between two vertices and is the length of shortest path between and in , if exists otherwise, . The open -neighbourhood set of vertex is the set of all vertices of which are different from and at distance at most from in , that is, . The closed -neighbourhood set of is defined as . Obviously . The total graph of a graph is the graph whose vertex set is and two vertices are adjacent whenever they are either adjacent or incident in . The Shadow graph of a connected graph is obtained by taking two copies of , say and . Join each vertex in to the neighbours of corresponding vertex in . The middle graph of a graph is the graph whose vertex set is and in which two vertices are adjacent whenever either they are adjacent edges of or one is a vertex of and the other is an edge incident with it. For standard terminology and notations we rely upon Balakrishnan and Ranganathan [1] and Haynes et al. [2]. The concept of distance dominating set was initiated by Slater [3] while the term distance -dominating set was coined by Henning et al. [4]. For an integer , a is a -dominating set of if every vertex in is within distance from some vertex . That is, . The minimum cardinality among all -dominating sets of is called the -domination number of and it is denoted by . It is obvious that . A -dominating set of cardinality is called a -set. The distance domination in the context of spanning tree is discussed by Griggs and Hutchinson [5] while bounds on the distance two-domination number and the classes of graphs attaining these bounds are reported in the work of Sridharan et al. [6]. In [7] Topp and Volkmann have discussed distance -domination as -covering and characterized connected graphs of order with distance -domination ( -covering). Application of distance domination in Ad Hoc wireless networking is briefly discussed by Wu and Li [8]. More details and bibliographic references on distance -domination can be found in a survey paper by Henning [9]. 2. Some Definitions and Main Results Proposition 1 (see [9]). For , let be a -dominating set of a graph . Then is a minimal -dominating
Subset-Sum Representations of Domination Polynomials  [PDF]
Tomer Kotek,James Preen,Peter Tittmann
Mathematics , 2012,
Abstract: The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the edge and vertex set of G. One of our main results is a representation of D(G,x) as a sum ranging over spanning bipartite subgraphs of G. We call a graph G conformal if all of its components are of even order. We show that the number of dominating sets of G equals a sum ranging over vertex-induced conformal subgraphs of G.
-Tuple Total Restrained Domination in Complementary Prisms  [PDF]
Adel P. Kazemi
ISRN Combinatorics , 2013, DOI: 10.1155/2013/984549
Abstract: In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of?? is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of?? . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [ -tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph. 1. Introduction Let be a simple graph with the vertex set?? and the edge set . The order?? and size?? of are denoted by and , respectively. The open neighborhood and the closed neighborhood of a vertex are and , respectively. Also the open neighborhood and the closed neighborhood of a subset are and , respectively. The degree of a vertex is . The minimum and maximum degree of a graph are denoted by and , respectively. If every vertex of has degree , then is called -regular. We write , , and for a complete graph, a cycle and a path of order , respectively, while denotes a complete -partite graph. The complement of a graph is denoted by and is a graph with the vertex set and for every two vertices and , if and only if . For each integer , the -join?? of a graph to a graph of order at least is the graph obtained from the disjoint union of and by joining each vertex of to at least vertices of [1]. Also, denotes the -join such that each vertex of is joined to exact vertices of . The complementary prism?? of is the graph formed from the disjoint union of and by adding the edges of a perfect matching between the corresponding vertices (same label) of and [2]. For example, the graph is the Petersen graph. Also, if , the graph is the corona , where the corona?? of a graph is the graph obtained from by attaching a pendant edge to each vertex of . The research of domination in graphs is an evergreen area of graph theory. Its basic concept is the dominating set. The literature on this subject has been surveyed and detailed in the two books by Haynes et al. [3, 4]. And many variants of the dominating set were introduced and the corresponding numerical invariants were defined for them. For example, the -tuple total dominating set is defined in [1] by Henning and Kazemi, which is an extension of the total dominating set (for more information see [5, 6]). Definition 1 (see [1]). Let be an integer and let be a graph with . A subset is called a -tuple total dominating set, briefly kTDS, in , if for each , . The minimum number of vertices of a -tuple
Some New Results on Global Dominating Sets  [PDF]
S. K. Vaidya,R. M. Pandit
ISRN Discrete Mathematics , 2012, DOI: 10.5402/2012/852129
Abstract: A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic. 1. Introduction We begin with finite and undirected simple graph of order . The set of vertices in a graph is called a dominating set if every vertex is either an element of or is adjacent to an element of . A dominating set is a minimal dominating set (MDS) if no proper subset is a dominating set. The minimum cardinality of a dominating set of is called the domination number of which is denoted by and the corresponding dominating set is called a -set of . The open neighborhood of is the set of vertices adjacent to and the closed neighborhood of is the set . The complement of is the graph with vertex set and two vertices are adjacent in if and only if they are not adjacent in . A subset is called a global dominating set in if is a dominating set of both and . The global domination number is the minimum cardinality of a global dominating set in . The concept of global domination in graph was introduced by Sampathkumar [1]. The upper bounds of global domination number are investigated by Brigham and Dutton [2] as well as by Poghosyan and Zverovich [3], while the global domination number of Boolean function graph is studied by Janakiraman et al. [4]. The global domination decision problems are NP-complete as discussed by Carrington [5] and by Carrington and Brigham [6]. The edge addition stable property in the context of global domination and connected global domination for cycle and path is discussed by Kavitha and David [7]. The concept of total global dominating set was introduced by Kulli and Janakiram [8] and they have also characterized total global dominating sets. The wheel is defined to be the join . The vertex corresponding to is known as apex vertex and the vertices corresponding to cycle are known as rim vertices. A shell graph is the graph obtained by taking concurrent chords in a cycle . The vertex at which all the chords are concurrent is called the apex. The shell graph is also called fan that is, . Definition 1.1. The one-point union of cycles of length denoted by is the graph obtained by identifying one vertex of each cycle. The one-point union of cycles is known as friendship graph which is denoted by . Definition 1.2 (see Shee and Ho [9]). Let be a graph and let , be copies of a graph .
On global location-domination in bipartite graphs  [PDF]
Carmen Hernando,Merce Mora,Ignacio M. Pelayo
Mathematics , 2015,
Abstract: A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the \emph{location-domination number} $\lambda(G)$. An LD-set $S$ of a graph $G$ is \emph{global} if it is an LD-set of both $G$ and its complement $\overline{G}$. The \emph{global location-domination number} $\lambda_g(G)$ is the minimum cardinality of a global LD-set of $G$. For any LD-set $S$ of a given graph $G$, the so-called \emph{S-associated graph} $G^S$ is introduced. This edge-labeled bipartite graph turns out to be very helpful to approach the study of LD-sets in graphs, particularly when $G$ is bipartite. This paper is mainly devoted to the study of relationships between global LD-sets, LD-codes and the location-domination number in a graph $G$ and its complement $\overline{G}$, when $G$ is bipartite.
Domination game: effect of edge- and vertex-removal  [PDF]
Bo?tjan Bre?ar,Paul Dorbec,Sandi Klav?ar,Ga?per Ko?mrlj
Mathematics , 2013,
Abstract: The domination game is played on a graph $G$ by two players, named Dominator and Staller. They alternatively select vertices of $G$ such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator's goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number $\gamma_g(G)$, and the Staller-start game domination number $\gamma_g'(G)$, is the number of vertices chosen in Game 1 and Game 2, respectively. It is proved that if $e\in E(G)$, then $|\gamma_g(G) - \gamma_g(G-e)| \le 2$ and $|\gamma_g'(G) - \gamma_g'(G-e)| \le 2$, and that each of the possibilities here is realizable by connected graphs $G$ for all values of $\gamma_g(G)$ and $\gamma_g'(G)$ larger than 5. For the remaining small values it is either proved that realizations are not possible or realizing examples are provided. It is also proved that if $v\in V(G)$, then $\gamma_g(G) - \gamma_g(G-v) \le 2$ and $\gamma_g'(G) - \gamma_g'(G-v) \le 2$. Possibilities here are again realizable by connected graphs $G$ in almost all the cases, the exceptional values are treated similarly as in the edge-removal case.
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