Average covering number for some graphs

2019 ◽  
Vol 53 (1) ◽  
pp. 261-268
Author(s):  
D. Doğan Durgun ◽  
Ali Bagatarhan
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Cartesian Product ◽  
Covering Number ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Covering Set ◽  
Local Covering

The interconnection networks are modeled by means of graphs to determine the reliability and vulnerability. There are lots of parameters that are used to determine vulnerability. The average covering number is one of them which is denoted by $ \overline{\beta }(G)$, where G is simple, connected and undirected graph of order n ≥ 2. In a graph G = (V(G), E(G)) a subset $ {S}_v\subseteq V(G)$ of vertices is called a cover set of G with respect to v or a local covering set of vertex v, if each edge of the graph is incident to at least one vertex of Sv. The local covering number with respect to v is the minimum cardinality of among the Sv sets and denoted by βv. The average covering number of a graph G is defined as β̅(G) = 1/|v(G)| ∑ν∈v(G)βν In this paper, the average covering numbers of kth power of a cycle $ {C}_n^k$ and Pn □ Pm, Pn □ Cm, cartesian product of Pn and Pm, cartesian product of Pn and Cm are given, respectively.

scholarly journals Graphs with equal domination and covering numbers

2019 ◽  
Vol 39 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Andrzej Lingas ◽  
Mateusz Miotk ◽  
Jerzy Topp ◽  
Paweł Żyliński
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Intersection Points ◽  
Covering Set ◽  
Appl Math ◽  
Side Result

Abstract A dominating set of a graph G is a set $$D\subseteq V_G$$D⊆VG such that every vertex in $$V_G-D$$VG-D is adjacent to at least one vertex in D, and the domination number $$\gamma (G)$$γ(G) of G is the minimum cardinality of a dominating set of G. A set $$C\subseteq V_G$$C⊆VG is a covering set of G if every edge of G has at least one vertex in C. The covering number $$\beta (G)$$β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which $$\gamma (G)=\beta (G)$$γ(G)=β(G) is denoted by $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β, whereas $${\mathcal {B}}$$B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β and $${\mathcal {B}}$$B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to $${\mathcal {B}}$$B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859–1867, 2013) allows to recognize all the graphs belonging to the set $${\mathcal {C}}_{\gamma =\beta }$$Cγ=β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in $$O(n \log n + m)$$O(nlogn+m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

scholarly journals On Topologies Induced by Graphs Under Some Unary and Binary Operations

2019 ◽  
Vol 12 (2) ◽  
pp. 499-505
Author(s):  
Caen Grace Sarona Nianga ◽  
Sergio R. Canoy Jr.
Keyword(s):  
Undirected Graph ◽  
Cartesian Product ◽  
Binary Operations ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺^2(𝑣) = {u ϵ V(G):  𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G^2[A] = V(G) \ N_G^2 [A] where N_G^2 [A] = A ∪ {v ϵ V(G):  𝑁_𝐺^2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

2021 ◽  
pp. 1-12
Author(s):  
Yuxing Yang
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Text Structure ◽  
Multiprocessor Systems ◽  
Cube Formula

Let [Formula: see text] be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super [Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

Complementary equitably totally disconnected equitable domination in graphs

2020 ◽  
pp. 2150043
Author(s):  
P. Nataraj ◽  
R. Sundareswaran ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Totally Disconnected

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

A Note of Independent Number and Domination Number of Qn,k,m-Graph

2019 ◽  
Vol 29 (03) ◽  
pp. 1950011
Author(s):  
Jiafei Liu ◽  
Shuming Zhou ◽  
Zhendong Gu ◽  
Yihong Wang ◽  
Qianru Zhou
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Domination Number ◽  
Multiprocessor System ◽  
Star Graph ◽  
Minimum Cardinality ◽  
Arrangement Graph ◽  
Independent Number ◽  
Potential Applications ◽  
By Products

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

On the fixed-point type Sylvester matrix equations over complete commutative dioids

2014 ◽  
Vol 27 ◽  
Author(s):  
Benham Hashemi ◽  
Mahtab Mirzaei Khalilabadi ◽  
Hanieh Tavakolipour
Keyword(s):  
Fixed Point ◽  
Tensor Product ◽  
Cartesian Product ◽  
Matrix Equations ◽  
Minimum Cardinality ◽  
Sylvester Matrix ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Sylvester Matrix Equations ◽  
Point Type

This paper extends the concept of tropical tensor product defined by Butkovic and Fiedler to general idempotent dioids. Then, it proposes an algorithm in order to solve the fixed-point type Sylvester matrix equations of the form X = A ⊗ X ⊕ X ⊗ B ⊕ C. An application is discussed in efficiently solving the minimum cardinality path problem in Cartesian product graphs.

On the minimum vertex covering transversal dominating sets in graphs and their classification

2017 ◽  
Vol 09 (05) ◽  
pp. 1750069 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Covering ◽  
Number Formula ◽  
Covering Set ◽  
Simple Connected Graph

Let [Formula: see text] be a simple and connected graph. A dominating set [Formula: see text] is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of [Formula: see text]. The vertex covering transversal domination number [Formula: see text] is the minimum cardinality among all vertex covering transversal dominating sets of [Formula: see text]. A vertex covering transversal dominating set of minimum cardinality [Formula: see text] is called a minimum vertex covering transversal dominating set or simply a [Formula: see text]-set. In this paper, we prove some general theorems on the vertex covering transversal domination number of a simple connected graph. We also provide some results about [Formula: see text]-sets and try to classify those sets based on their intersection with the minimum vertex covering sets.

On the Hamilton Cycle of the Hypercube

2011 ◽  
Vol 480-481 ◽  
pp. 922-927 ◽  
Author(s):  
Yan Zhong Hu ◽  
Hua Dong Wang
Keyword(s):  
Euler Characteristic ◽  
Interconnection Networks ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
The Other ◽  
Product Graph ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Isomorphic Graphs ◽  
The Relationship

Hypercube is one of the basic types of interconnection networks. In this paper, we use the concept of the Cartesian product graph to define the hypercube Qn, we study the relationship between the isomorphic graphs and the Cartesian product graphs, and we get the result that there exists a Hamilton cycle in the hypercube Qn. Meanwhile, the other properties of the hypercube Qn, such as Euler characteristic and bipartite characteristic are also introduced.

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