A Study On Very Strongly Perfect And Trivially Perfect Graphs

Abstract

The first use of the phrase perfect graph appears to be in newlinea 1963 paper by C. Berge, after whom Berge graphs are named. In his newlinepaper, he conjectured the equivalence of the perfect graph and Berge newlinegraph definitions; Berge s conjecture was later proven as the strong newlineperfect graph theorem. newlineIn many cases, new classes of perfect graphs that have been newlineintroduced were motivated by generalizing known classes of perfect newlinegraphs. Many classes of perfect graphs are, therefore, subclasses of newlineother classes of perfect graphs. In this line of thought, new classes of newlinegraphs named Very Strongly Perfect Graph and Trivially Perfect newlineGraph are investigated. It is given a characterization of the very strong newlineand trivially perfect graph. Very strongly perfect structures of both newlinebipartite and perfectly orderable graphs are discussed. Throughout the newlinethesis, products on connected simple graphs and#119866;1 and and#119866;2 with no induced newlineand#119862;2and#119896;+1 for and#119896; and#8805; 2 are considered. Let and#119866;1and#9649; and#119866;2 be the Cartesian product of newlinesimple, connected, and finite graph and#119866;1 and and#119866;2. It is given the necessary newlineand sufficient conditions for the Cartesian product of graphs to be VSP. newlineFurther, co-strongly perfect graphs are characterized. The structural newlineproperties of VSPG, odd cycles, perfectly orderable, bipartite, and newlinex newlinestrongly perfect graphs are discussed. It is given an algorithm for the newlinestrong independent set on and#119862;2and#119896;+1 or and#119862;2and#119896;+1 + and#119890;, and#119896; and#8805; 2 free graphs. These newlinemethods can be used to determine one of the best mathematical models newlinefor a real situation, where one would like to choose an optimal set of newlineleaders from a given set of people. In the future, this investigation will newlinebe more applicable to the remaining graph classes also. Further, is newlineextended our work to introduce the residual division graph of the lattice newlinemodule and characterized the co-multiplication lattice module and prime newlinelattice module. Also, it is characterized by the vertex set of residual newlinedivision graphs of the multiplication lattice module. It is found that the newlineresidual division graph of an interval lattice