A Study on Restrained Geodetic Domination in Graphs

Title

A Study on Restrained Geodetic Domination in Graphs

Creator

Joy, John.

Contributor

Shathish, Sangeetha.

Description

In a graph G = (V, E), the shortest path between any two vertices u and v in G is u and#8722; v geodesic. This distance concept leads to the introduction of geodetic set and geodetic number which has wide applications in location theory and convexity theory. A vertex subset S of a graph G is said to be a geodetic set, if all vertex in G is in u and#8722; v geodesic for some pair of vertices u and v in S. The minimum cardinality of such a set is the geodetic number and is denoted as g(G). A vertex subset M of a graph G is said to be a dominating set of G if for all vertex v and#8712; V (G), either v and#8712; M or v is adjacent to a vertex in M. The minimum cardinality of such a set is the domination number and is denoted by and#947;(G). In general, the geodetic set and newlinethe dominating set of a graph need not be the same. This led to the study of the geodetic dominating set. If a geodetic set S is a dominating set of a graph G, then S is called a geodetic dominating set. The minimum cardinality of such a set is the geodetic domination number, which is represented by and#947;g(G). There are several studies done on the geodetic and domination concepts so far. In the present study, we have explored the concept of restrained geodetic domination and its structural properties in graphs particularly in product graphs and derived graphs. A vertex subset S of a graph G = (V, E) is called a restrained geodetic dominating set if S is a geodetic dominating set of G and lt V and#8722; S gt has no isolated vertex. The minimum cardinality of such a set is called restrained geodetic domination number, which is denoted by and#947;gr(G). We have studied this concept for diand#64256;erent classes of graphs and concerning the graph operations such as Cartesian product, corona product, and join of graphs. Further, the study is extended to restrained geodetic domination in derived graphs such as edge subdivision graph, line graph and power of a graph. Also, investigated the properties of graphs with the restrained geodetic domination number equal to the order of the graph.

Source

Author's Submission

Date

2023-01-01

Publisher

Christ(Deemed to be University)

Subject

Mathematics and Statistics

Rights

Open Access

Relation

61000226

Format

PDF

Language

English

Type

PhD

Identifier

http://hdl.handle.net/10603/492346