IJPAM: Volume 119, No. 1 (2018)

Title

THE DETOUR IRREDUNDANT NUMBER OF A GRAPH

Authors

S. Delbin Prema$^1$, C. Jayasekaran$^2$
$^1$Department of Mathematics
RVS Technical Campus - Coimbatore
Coimbatore, 641402, Tamil Nadu, INDIA
$^2$Department of Mathematics
Pioneer Kumaraswamy College
Nagercoil, 629003, Tamil Nadu, INDIA

Abstract

For two vertices $u$ and $v$ of a connected graph $G$, the set $I_D[u, v]$ consists of all those vertices lying on $u-v$ detour in $G$. Given a set $S$ of vertices of $G$, the union of all sets $I_D[u, v]$ for $u, v \in S$ is denoted by $I_D[S]$. A detour convex set $S$ satisfies $I_D[S] = S$. The detour convex hull $[S]_D$ of $S$ is the smallest detour convex set containing $S$. The detour hull number $d_h(G)$ is the minimum cardinality among the subsets $S$ of $V$ with $[S]_D = V$. In this paper, we introduce and study the detour irredundant number of a graph. A set $S$ of vertices of $G$ is a detour irredundant set if $u \notin I_D[S -\{u\}]$ for all $u \in S$ and the maximum cardinality of a detour irredudant set is its detour irredundant number $dir(G)$ of $G$. We determine the detour irredundant number of certain standard classes of graphs. Certain general properties of these concepts are studied. We characterize the classes of graphs of order $n$ for which $dir(G) = n$ or $dir(G) = n-1$ or $dir(G) = n - 2$, respectvely. We prove that for any integers $a$ and $b$ with $2 \leq a \leq b$, there exists a connected graph $G$ such that $d_h(G)=a$ and $dir(G)=b$. We also prove that for integers $a,b$ and $k\geq 2$ with $a< b\leq 2a$, there exists a connected graph $G$ with $rad_D(G)=a, diam_D(G)=b$ and $dir(G)=k$.

History

Received: March 20, 2017
Revised: June 5, 2018
Published: June 5, 2018

AMS Classification, Key Words

AMS Subject Classification: 05C12
Key Words and Phrases: detour convex set, extreme vertex, detour hull number, detour irredundant sets, detour irredundant number

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Bibliography

1
F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, ReadingMA, (1990).

2
G. Chartrand and P. Zhang, Introduction to Graph Theory, Tata McGraw- Hill Edition, New Delhi,2006.

How to Cite?

DOI: 10.12732/ijpam.v119i1.4 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 1
Pages: 53 - 62


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