International Journal of Advances in Mathematics
International Journal of Advances in Mathematics
2456-6098
Volume 2018
Number 3
2018
May
01
Total resolving number of subdivision and total graphs
34
40
EN
Shunmugapriya
N.
Department of Mathematics,
Manonmaniam Sundaranar University, Tirunelveli - 627 012, Tamil Nadu, India.
nshunmugapriya2013@gmail.com
Paulraj Joseph
J.
Department of Mathematics,
Manonmaniam Sundaranar University, Tirunelveli - 627 012, Tamil Nadu, India.
prof.jpaulraj@gmail.com
Let $G = (V, E)$ be a simple connected graph. An ordered subset $W$ of $V$ is said to be a \textit{resolving set} of $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W.$ The minimum cardinality of a resolving set is called the \textit{resolving number} of $G$ and is denoted by $r(G).$ As an extension, the \textit{total resolving number} was introduced in \cite{F} as the minimum cardinality taken over all resolving sets in which $\left\langle W \right\rangle$ has no isolates and it is denoted by $tr(G).$ In this paper, we obtain the bounds on the total resolving number of subdivision graphs and total graphs. Also, we characterize the extremal graphs.
Resolving number, total resolving number, subdivision graph and total graph.
http://adv-math.com/total-resolving-number-subdivision-total-graphs/
http://adv-math.com/wp-content/uploads/2018/05/201802.pdf