The Transversal Neighborhood Domination Number on Parachute Graph and Semi-Parachute Graph

Abstract

Given V be a set of vertices on a graph G. A set D ⊆ V, is dominating set of G = (V, E) if all the vertex that is not in the set D are neighbors to at least one vertex of D. The smallest number of elements in D is known as the domination number. A set D ⊆ V(G) is called a transversal neighborhood-dominating set if D is the dominating set of G and intersects with every minimum neighborhood set. The transversal neighborhood domination number of G is the smallest number of elements in each transversal neighborhood-dominating set.. In this paper, we discuss transversal neighborhood domination number on a parachute graph and a semi-parachute graph. The transversal neighborhood domination number on parachute graph is (n + 3)/3 for every n = 3 k, (n + 5)/3 for every n = 3 k + 1, and (n + 4)/3 for every n = 3 k + 2. The neighborhood transversal domination number on the semi-parachute graph is (n + 1)/2 for every n = 2 k + 1 and (n + 2)/2 for every n = 2 k.