The near geodetic number of a graph.
Saved in:
| Title: | The near geodetic number of a graph. |
|---|---|
| Authors: | Lenin, R.1, Kathiresan, KM.1 |
| Source: | Journal of Discrete Mathematical Sciences & Cryptography. Aug2017, Vol. 20 Issue 5, p1091-1100. 10p. |
| Subjects: | Geodesics, Graph theory, Geometric vertices |
| Abstract: | The intervalI(u, v) for any two vertices consists of all those vertices lying on au – v geodesic(shortest path) inG. IfSis a set of vertices ofG,thenI(S) is the union of all setsI(u, v) foru, v∈S.The geodetic numberg(G) is the minimum cardinality among the subsetsSofV(G) withI(S) =V(G). A set S ⊇V(G) is a near geodetic set if for everyvinV(G) –S, there exists somex, yinSwith |I(x,y) ∩N(v)|≥ 2. The near geodetic numbergnear(G) is the minimum cardinality of a near geodetic set inG.In this paper, we proved that ifGis a graph of ordernand clique numberw(G), thengnear(G) ≤n− ω(G) + 2 and this bound is sharp. Further we proved that any positive integersa, bandn, with 2 ≤a |
| Copyright of Journal of Discrete Mathematical Sciences & Cryptography is the property of Taru Publications and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| Abstract: | The intervalI(u, v) for any two vertices consists of all those vertices lying on au – v geodesic(shortest path) inG. IfSis a set of vertices ofG,thenI(S) is the union of all setsI(u, v) foru, v∈S.The geodetic numberg(G) is the minimum cardinality among the subsetsSofV(G) withI(S) =V(G). A set S ⊇V(G) is a near geodetic set if for everyvinV(G) –S, there exists somex, yinSwith |I(x,y) ∩N(v)|≥ 2. The near geodetic numbergnear(G) is the minimum cardinality of a near geodetic set inG.In this paper, we proved that ifGis a graph of ordernand clique numberw(G), thengnear(G) ≤n− ω(G) + 2 and this bound is sharp. Further we proved that any positive integersa, bandn, with 2 ≤a<b≤n− 2, there exists a connected graphGof ordernwithg(G) =aandgnear(G) =b.Also we shown that, for positive integersn,k,lwithn−k−l+ 2 ≥ 0, 2 ≤k<n− 1 and 2 ≤l≤n, there exist a connected graphGof ordernwith clique numberω(G) = 1 andgnear(G) =K. [ABSTRACT FROM AUTHOR] |
|---|---|
| ISSN: | 09720529 |
| DOI: | 10.1080/09720529.2016.1248689 |
