scholarly journals The 3-Good-Neighbor Connectivity of Modified Bubble-Sort Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Yanling Wang ◽  
Shiying Wang
Keyword(s):  
Cayley Graph ◽  
Connected Graph ◽  
Minimum Cardinality ◽  
Neighbor Connectivity ◽  
Good Neighbor

Let G = V G , E G be a connected graph. A subset F ⊆ V G is called a g -good-neighbor cut if G − F is disconnected and each vertex of G − F has at least g neighbors. The g -good-neighbor connectivity of G is the minimum cardinality of g -good-neighbor cuts. The n -dimensional modified bubble-sort graph MB n is a special Cayley graph. It has many good properties. In this paper, we prove that the 3-good-neighbor connectivity of MB n is 8 n − 24 for n ≥ 6 .

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Hussain
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Four Dimensions ◽  
Symmetric Graphs ◽  
Metric Dimensions ◽  
Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

scholarly journals Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 300 ◽  
Author(s):  
Zafar Hussain ◽  
Mobeen Munir ◽  
Maqbool Chaudhary ◽  
Shin Kang
Keyword(s):  
Electronic Structure ◽  
Transport Properties ◽  
Present Article ◽  
Connected Graph ◽  
Metric Dimension ◽  
Lattice Structures ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Mathematical Sciences ◽  
Metric Basis

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

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.

THE FORCING EDGE FIXING EDGE-TO-VERTEX MONOPHONIC NUMBER OF A GRAPH

2013 ◽  
Vol 05 (04) ◽  
pp. 1350034
Author(s):  
J. JOHN ◽  
K. UMA SAMUNDESVARI
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
General Properties

For a connected graph G = (V, E), a set Se ⊆ E(G)–{e} is called an edge fixing edge-to-vertex monophonic set of an edge e of a connected graph G if every vertex of G lies on an e – f edge-to-vertex monophonic path of G, where f ∈ Se. The edge fixing edge-to-vertex monophonic number mefev(G) of G is the minimum cardinality of its edge fixing edge-to-vertex monophonic sets of an edge e of G. A subset Me ⊆ Se in a connected graph G is called a forcing subset for Se, if Se is the unique edge fixing edge-to-vertex monophonic set of e of G containing Me. A forcing subset for Se of minimum cardinality is a minimum subset of Se. The forcing edge fixing edge-to-vertex monophonic number of G denoted by fefev(G) = min {fefev(Se)}, where the minimum is taken over all cardinality of a minimal edge fixing edge-to-vertex monophonic set of e of G. The forcing edge fixing edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. It is shown that for every integers a and b with 0 ≤ a ≤ b, b ≥ 1, there exists a connected graph G such that fefev(G) = a, mefev(G) = b.

scholarly journals Domination of Cayley Digraph and its Complement

2019 ◽  
Vol 8 (2S11) ◽  
pp. 4005-4008
Keyword(s):  
Cayley Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Generating Set ◽  
Cayley Digraph

A Cayley graph constructed out of a group Γ and its generating set A is denoted by Cay (Γ, A). The digraph with the same node set as the original digraph is said to be a complement digraph if it has an edge from x to y exactly when the original digraph does not have an edge from x to y. A subset Ɖ of V is called a dominating set if each vertex in V- Ɖ is adjacent to at least one vertex in Ɖ. The minimum cardinality of a dominating set is called Domination number which is denoted by γ. The domination number of Cayley digraphs and Complement of Cayley digraphs of groups are investigated in this paper. Also, the graph relationship involving domination parameters in a graph and its complement are studied.

scholarly journals The Partition Dimension of a Path Graph

2021 ◽  
Vol 13 (2) ◽  
pp. 66
Author(s):  
Vivi Ramdhani ◽  
Fathur Rahmi
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Path Graph ◽  
Partition Dimension

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph,  with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

scholarly journals Identifying Codes of Lexicographic Product of Graphs

10.37236/2974 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Min Feng ◽  
Min Xu ◽  
Kaishun Wang
Keyword(s):  
Connected Graph ◽  
Necessary Condition ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Two Parameters ◽  
Sufficient And Necessary Condition ◽  
Product Of Graphs

Let $G$ be a connected graph and $H$ be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product $G[H]$ of $G$ and $H$. We first introduce two parameters of $H$, which are closely related to identifying codes of $H$. Then we provide the sufficient and necessary condition for $G[H]$ to be identifiable. Finally, if $G[H]$ is identifiable, we determine the minimum cardinality of identifying codes of $G[H]$ in terms of the order of $G$ and these two parameters of $H$.

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