Interval graphs, their importance over the years can be seen in the increasing number of researchers trying to explore the field. In this paper we presented the comparison of degree of complementary dominating set of dominating set using Euclidean division algorithm with divisor 5 for interval graph G corresponding to an interval family I. In particular we prove a necessary condition under which the complementary dominating set as well as the domination number.
Graph theory is the study of graphs, which are mathematical structures used to demonstrate pair wise relations between objects. A dominating set is used as a backbone for communication. Among various applications of the theory of domination, chromatic number and total domination the most often discussed is a communication network. The Circular-arc graphs have drawn the attention of research for over number of years. They are extensively studied and revealed their practical relevance for modeling problems arising in the real world. The aim of this paper is to find the domination numbers, Chromatic numbers and Total domination numbers of all powers of cycles using a Circular-arc graph G corresponding to a Circular-arc family A.
Domination in graphs has applications to several fields. A similar problem occurs when the maximum distance to a facility is fixed and one attempts to minimize the no of facilities necessary so that everyone is serviced. The prominence of Circular-arc graphs in the current times is evident from the enhanced research focus and attempts for extensive exploration during the recent years. In this paper, we derived the some properties on sum of degrees of elements in a minimum dominating set through the quotient of divisor 3, domination number and the number of vertices of the circular-arc graphs corresponding to a circular-arc family in a particular case of each arc intersects the next arc only by using Euclidean division lemma.
coloring takes its name from the map coloring application, we assign labels to vertices. When the numerical value of the labels is unimportant, we call them colors to indicate that they may be elements of any set. In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths. The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this paper we show that the rainbow connection number of an interval graph, which are of the form the rainbow connection number is less than or equal to the diameter of the graph G plus one.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.