Perfect Geodomination in Graphs
DoostAli Mojdeh
Department of Mathematics,
University of Azad Islamic Noor Branch,
Noor, Iran
Abstract
A pair x,y of vertices in a
nontrivial connected graph G is said to geodominate a vertex v
of G if either v
{x,y} or v lies in an x-y geodesic
of G. A set S of vertices of G is a geodominating set if
every vertex of G is geodominated by some pair of vertices of
S. A vertex of G is link-complete if the subgraph induced by
its neighborhood is complete. A perfect geodominating set is a
geodominating set S such that any vertex vV(G)\S
is geodominated by exactly one pair of vertices of S. A
k-perfect geodominating set is a geodominating set S such that
any vertex vV(G)\S is geodominated by exactly one
pair x, y of vertices of S with d(x, y) = k. The cardinality
of a minimum perfect geodominating set in G is its perfect
geodomination number gp(G) and the cardinality of a minimum
k-perfect geodominating set in G is its k-perfect
geodomination number gkp(G). We study perfect and k-perfect
geodomination numbers of a graph G.
