scholarly journals Efficient open domination in graph products

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael Gonzalez Yero
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Graph Products ◽  
International Audience ◽  
Domination Graphs

Graph Theory International audience A graph G is an efficient open domination graph if there exists a subset D of V(G) for which the open neighborhoods centered in vertices of D form a partition of V(G). We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is K2.

scholarly journals On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Olivier Baudon ◽  
Julien Bensmail ◽  
Rafał Kalinowski ◽  
Antoni Marczyk ◽  
Jakub Przybyło ◽  
...  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Vertex Set ◽  
International Audience ◽  
Positive Integers

Graph Theory International audience A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

scholarly journals The generalized 3-connectivity of Cartesian product graphs

2012 ◽  
Vol Vol. 14 no. 1 (Graph Theory) ◽  
Author(s):  
Hengzhe Li ◽  
Xueliang Li ◽  
Yuefang Sun
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Cartesian Product ◽  
Vertex Connectivity ◽  
Connected Graphs ◽  
Generalized Connectivity ◽  
Product Graphs ◽  
International Audience ◽  
Cartesian Product Graphs ◽  
Internally Disjoint Trees

Graph Theory International audience The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 <= i, j <= r. For an integer k with 2 <= k <= n, the k-connectivity kappa(k) (G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.

scholarly journals On the 1-2-3-conjecture

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Akbar Davoodi ◽  
Behnaz Omoomi
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Cartesian Product Of Graphs ◽  
Edge Weighting ◽  
International Audience ◽  
Product Of Graphs ◽  
Proper Vertex

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

scholarly journals The generalized 3-connectivity of Lexicographic product graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Xueliang Li ◽  
Yaping Mao
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Natural Generalization ◽  
Upper Bounds ◽  
Lexicographic Product ◽  
Vertex Connectivity ◽  
Connected Graphs ◽  
Product Graphs ◽  
International Audience ◽  
Lexicographic Product Graphs

Graph Theory International audience The generalized k-connectivity κk(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G^H and G&Box;H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G^H)&#x2265; κ3(G)|V(H)|. We also give upper bounds for κ3(G&Box; H) and κ3(G^H). Moreover, all the bounds are sharp.

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 p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals The Second Hyper-Zagreb Index of Complement Graphs and Its Applications of Some Nano Structures

2021 ◽  
pp. 54-75
Author(s):  
Mohammed Alsharafi ◽  
Yusuf Zeren ◽  
Abdu Alameri
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Quantitative Structure Activity Relationship ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graph Theory ◽  
Mathematical Operations

In chemical graph theory, a topological descriptor is a numerical quantity that is based on the chemical structure of underlying chemical compound. Topological indices play an important role in chemical graph theory especially in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In this paper, we present explicit formulae for some basic mathematical operations for the second hyper-Zagreb index of complement graph containing the join G1 + G2, tensor product G1 \(\otimes\) G2, Cartesian product G1 x G2, composition G1 \(\circ\) G2, strong product G1 * G2, disjunction G1 V G2 and symmetric difference G1 \(\oplus\) G2. Moreover, we studied the second hyper-Zagreb index for some certain important physicochemical structures such as molecular complement graphs of V-Phenylenic Nanotube V PHX[q, p], V-Phenylenic Nanotorus V PHY [m, n] and Titania Nanotubes TiO2.

scholarly journals Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

2020 ◽  
Vol 1 (2) ◽  
pp. 64
Author(s):  
Nurma Ariska Sutardji ◽  
Liliek Susilowati ◽  
Utami Dyah Purwati
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Metric Dimension ◽  
Star Graph ◽  
Product Graph ◽  
Path Graph ◽  
Strong Metric Dimension ◽  
Metric Dimensions ◽  
Development Result ◽  
Corona Product

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

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