scholarly journals Hyperbolicity and Chordality of a Graph

10.37236/530 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yaokun Wu ◽  
Chengpeng Zhang
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Chordal Graph ◽  
Isometric Subgraph

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ by at most $2\delta.$ The graph $G$ is $k$-chordal provided it has no induced cycle of length greater than $k.$ Brinkmann, Koolen and Moulton find that every $3$-chordal graph is $1$-hyperbolic and that graph is not $\frac{1}{2}$-hyperbolic if and only if it contains one of two special graphs as an isometric subgraph. For every $k\geq 4,$ we show that a $k$-chordal graph must be $\frac{\lfloor\frac{k}{2}\rfloor}{2}$-hyperbolic and there does exist a $k$-chordal graph which is not $\frac{\lfloor \frac{k-2}{2}\rfloor}{2}$-hyperbolic. Moreover, we prove that a $5$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does not contain any of a list of five special graphs as an isometric subgraph.

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

A Note on the Diameter of a Graph

1968 ◽  
Vol 11 (3) ◽  
pp. 499-501 ◽  
Author(s):  
J. A. Bondy
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Distance ◽  
Image Position

The distance d(x, y) between vertices x, y of a graph G is the length of the shortest path from x to y in G. The diameter δ(G) of G is the maximum distance between any pair of vertices in G. i.e. δ(G) = max max d(x, y). In this note we obtain an upper boundx ε G y ε Gfor δ(G) in terms of the numbers of vertices and edges in G. Using this bound it is then shown that for any complement-connected graph G with N verticeswhere is the complement of G.

scholarly journals On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

A Fast Algorithm for Constructing Sparse Euclidean Spanners

1997 ◽  
Vol 07 (04) ◽  
pp. 297-315 ◽  
Author(s):  
Gautam Das ◽  
Giri Narasimhan
Keyword(s):  
Greedy Algorithm ◽  
Shortest Path ◽  
Connected Graph ◽  
Fast Algorithm ◽  
Dimensional Space ◽  
Time Algorithm ◽  
Real Constant ◽  
Clustering Techniques ◽  
Positive Edge ◽  
The Greedy Algorithm

Let G = (V,E) be a n-vertex connected graph with positive edge weights, and let t > 1 be a real constant. A subgraph G' is a t-spanner if for all u,v ∊ V, the weight of the shortest path between u and v in G' is at most t times the weight of the corresponding shortest path in G. We design an O(n log 2 n) time algorithm which, given a set V of n points in k-dimensional space (for any fixed k), and any real constant t > 1, produces a t-spanner of the complete Euclidean graph of V. This algorithm retains the spirit of a recent O(n3 log n) time greedy algorithm which produces t-spanners; we use graph clustering techniques to achieve a more efficient implementation. Our spanners have O(n) edges and weight O(1)· wt(MST), which is similar to the size and weight of spanners constructed by the greedy algorithm. The constants implicit in the O-notation depend upon t and k.

CONSTRUCTING MULTIDIMENSIONAL SPANNER GRAPHS

1991 ◽  
Vol 01 (02) ◽  
pp. 99-107 ◽  
Author(s):  
JEFFERY S. SALOWE
Keyword(s):  
Shortest Path ◽  
Complete Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Input Parameter ◽  
Minimum Spanning Trees ◽  
Positive Edge ◽  
Spanning Subgraph ◽  
Vertex Set

Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in [Formula: see text] and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing [Formula: see text] edges in [Formula: see text] time. We apply this spanner to the construction of approximate minimum spanning trees.

scholarly journals On the Geodesic Identification of Vertices in Convex Plane Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Fawaz E. Alsaadi ◽  
Muhammad Salman ◽  
Masood Ur Rehman ◽  
Abdul Rauf Khan ◽  
Jinde Cao ◽  
...  
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Metric Dimension ◽  
Plane Graphs ◽  
Convex Plane ◽  
Strong Metric Dimension ◽  
Minimum Number

A shortest path between two vertices u and v in a connected graph G is a u − v geodesic. A vertex w of G performs the geodesic identification for the vertices in a pair u , v if either v belongs to a u − w geodesic or u belongs to a v − w geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in G is called the strong metric dimension of G . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

scholarly journals BI-DIMENSI METRIK DARI GRAF ANTIPRISMA

2020 ◽  
Vol 20 (2) ◽  
pp. 53
Author(s):  
Hendy Hendy ◽  
M. Ismail Marzuki
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality

Let G = (V, E) be a simple and connected graph. For each x ∈ V(G), it is associated with a vector pair (a, b), denoted by S x , corresponding to subset S = {s1 , s2 , ... , s k } ⊆ V(G), with a = (d(x, s1 ), d(x, s2 ), ... , d(x, s k )) and b = (δ(x, s1 ), δ(x, s2 ), ... , δ(x, s k )). d(v, s) is the length of shortest path from vertex v to s, and δ(v, s) is the length of the furthest path from vertex v to s. The set S is called the bi-resolving set in G if S x ≠ S y for any two distinct vertices x, y ∈ V(G). The bi- metric dimension of graph G, denoted by β b (G), is the minimum cardinality of the bi-resolving set in graph G. In this study we analyze bi-metric dimension in the antiprism graph (A n ). From the analysis that has been done, it is obtained the result that bi-metric dimension of graph A n , β b (A n ) is 3. Keywords: Antiprism graph, bi-metric dimension, bi-resolving set. .

scholarly journals On bounded partition dimension of different families of convex polytopes with pendant edges

2021 ◽  
Vol 7 (3) ◽  
pp. 4405-4415
Author(s):  
Adnan Khali ◽  
◽  
Sh. K Said Husain ◽  
Muhammad Faisal Nadeem ◽  
◽  
...  
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Convex Polytopes ◽  
General Family ◽  
Minimum Value ◽  
Partition Dimension ◽  
Simple Connected Graph

<abstract><p>Let $ \psi = (V, E) $ be a simple connected graph. The distance between $ \rho_1, \rho_2\in V(\psi) $ is the length of a shortest path between $ \rho_1 $ and $ \rho_2. $ Let $ \Gamma = \{\Gamma_1, \Gamma_2, \dots, \Gamma_j\} $ be an ordered partition of the vertices of $ \psi $. Let $ \rho_1\in V(\psi) $, and $ r(\rho_1|\Gamma) = \{d(\rho_1, \Gamma_1), d(\rho_1, \Gamma_2), \dots, d(\rho_1, \Gamma_j)\} $ be a $ j $-tuple. If the representation $ r(\rho_1|\Gamma) $ of every $ \rho_1\in V(\psi) $ w.r.t. $ \Gamma $ is unique then $ \Gamma $ is the resolving partition set of vertices of $ \psi $. The minimum value of $ j $ in the resolving partition set is known as partition dimension and written as $ pd(\psi). $ The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as $ R_n^P $, $ D_n^p $ and $ Q_n^p $ and proved that these graphs have bounded partition dimension.</p></abstract>

scholarly journals Distance-transitive graphs of valency five

1987 ◽  
Vol 30 (1) ◽  
pp. 73-81 ◽  
Author(s):  
A. Gardiner ◽  
Cheryl E. Praeger
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Transitive Graphs

If u and v are vertices of the (finite, connected) graph Γ, let d(u, v) denote the length of the shortest path joining u to v in Γ. The graph Γ is said to be distance-transitive if whenever d(u, v) = d(u′, v′), there exists an automorphism g of Γ such that ug = u′ and if vg = v′. Distance-transitive graphs of valency 3 and 4 were originally classified [2, 11, 12, 13] by using a computer to generate all “feasible intersection arrays” (cf. [1, Chapter 20]). In both cases a classification has since been given by hand [4, 5]. Wecontinue this latter tradition and prove the following theorem—which was recently proved independently by Ivanov et al. using a computer [10].

scholarly journals DIMENSI METRIK DAN DIAMETER DARI GRAF ULAT Cm,n

2017 ◽  
Vol 6 (1:) ◽  
pp. 57-62
Author(s):  
Restu Ria Wantika
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Maximum Distance ◽  
Star Graph ◽  
Minimum Cardinality ◽  
Graph Diameter ◽  
Metric Dimensions ◽  
Shaped Graph ◽  
Dimensional Graph

Graf is a pair (V,E) where V set of vertices is not empty and E set side. Let u and v are the vertices in a connected graph G, then the distance d (u, v) is the length of the shortest path between u and v in G. The diameter of graph G is the maximum distance of d (u, v) .For the set of ordered  of vertices in a connected graph G and vertex , the representation of v to W is . If  r (v│W) for each node v∈V (G) are different, then W is called the set of variants from G and the minimum cardinality of the set differentiator is referred to as the metric dimensions. Based on the characteristics of the vertices and sides of the graph have many types of them are caterpillars and graph graph fireworks, which both have in common at the center of the graph shaped trajectory and earring star-shaped graph. In this paper will prove that Graf caterpillar with   has diameter  and metric dimensions . Keywords: dimensional graph, graph diameter, star graph, graph caterpillar ..  

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