TY - JOUR
ID - 5750
TI - The geodetic domination number for the product of graphs
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Chellathurai, S. Robinson
AU - Vijaya, S. Padma
AD - Scott Christian College
AD - University College of Engineering Nagercoil
Y1 - 2014
PY - 2014
VL - 3
IS - 4
SP - 19
EP - 30
KW - Cartesian product
KW - strong product
KW - geodetic number
KW - Domination Number
KW - geodetic domination number
DO - 10.22108/toc.2014.5750
N2 - A subset $S$ of vertices in a graph $G$ is called a geodetic set if every vertex not in $S$ lies on a shortest path between two vertices from $S$. A subset $D$ of vertices in $G$ is called dominating set if every vertex not in $D$ has at least one neighbor in $D$. A geodetic dominating set $S$ is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number $g(G) (\gamma(G),\gamma_g(G))$ of $G$ is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in $G$. In this paper, we show that if a triangle free graph $G$ has minimum degree at least 2 and $g(G) = 2$, then $\gamma _g(G) = \gamma(G)$. It is shown, for every nontrivial connected graph $G$ with $\gamma(G) = 2$ and $diam(G) > 3$, that $\gamma_g(G) > g(G)$. The lower bound for the geodetic domination number of Cartesian product graphs is proved. Geodetic domination number of product of cycles (paths) are determined. In this work, we also determine some bounds and exact values of the geodetic domination number of strong product of graphs.
UR - https://toc.ui.ac.ir/article_5750.html
L1 - https://toc.ui.ac.ir/article_5750_87bdcf395fa6e3fd7e39a154bc0f1442.pdf
ER -