A study on domination eccentric domination in graphs and some operations on graphs

Abstract

Domination as a theoretical concept in graph theory was formalized by Berge and newlineOre. Ore introduced the word domination in his famous book Theory of graphs , newlinepublished in 1962. A set D and#8838; V is said to be a dominating set in G, if every vertex in newlineVand#8722;D is adjacent to some vertex in D. The minimum cardinality of a dominating set is newlinecalled the domination number and is denoted by and#947;(G). A dominating set with cardinality newlineand#947;(G) is known as minimum dominating set or a and#947;-set. newlineMost of additional variations of domination can be obtained by imposing newlinerestrictions on the dominating set or its complement. A dominating set D and#8838; V is a newlineconnected eccentric dominating set if for every v and#8712; Vand#8722;D, there exists at least one newlineeccentric vertex of v in D and ltDgt is connected. The minimum cardinality of a newlineconnected eccentric dominating set is called the connected eccentric domination number newlineand is denoted by and#947;ced(G). newlineIn this thesis, some bounds for connected eccentric domination number in trees newlineare obtained. Co-eccentric eccentric domination number and b-domination number in a newlineconnected graph G are defined and studied. Sharp bounds for these parameters are newlineobtained. Graphs with radius one for which and#947;cee(G) = 1, 2, nand#8722;2 and n are characterized. newlineAlso, for a tree T, upper and lower bounds of and#947;cee(T) is found out. Connected domatic newlinenumber, Co-eccentric eccentric domatic number and b-domatic number are defined and newlinestudied. Exact values of some particular graphs are given. newlineAlso, boundary graph (digraph), boundary neighbor graph (digraph), iterated newlineboundary digraph and iterated boundary neighbor digraph are defined. Further, Boolean newlinegraph BG(G) is investigated for some classes of graphs. Its structural properties, newlineeccentricity properties and traversability properties are examined. Characterizations and newlineedge decompositions of BG(G) are also discussed. newlineAlso some applications of the concepts introduced in this work are given.