Complementary tree nil domination number with chromatic number and connectivity of a graph

Abstract

Graph theory is a stand out applicable limb of advanced mathematics with its newlinemulti-pronged technological advancements. The first paper in graph theory was written newlineby Euler in 1736 when he settled the well-known unsolved problem of his day, known as newlinethe Konigsberg Bridge problem. Some puzzles and various problems of practical nature newlinehave been instrumental in the development of various topics in graph theory. Some of newlinethese areas are Coloring of Graphs, Matching Theory, Domination Theory, Labeling of newlineGraphs and areas related to Algebraic Graph Theory. De jaenisch discovered and studied newlinein depth the difficulties related to (n × n) chessboard problem that how many number of newlinequeens are required to lead. This seems the historical roots dating back to 1862 of this newlinesubject. The beginning of extensive study of dominating sets took place in nearly 1960. newlineBerge elaborated the conception of the domination number of a graph which is known as newlinecoefficient External Stability, in 1962. For the same concept, Ore provided the name newline Dominating set and Domination number . After this, many theories were revealed on newlinegraph theory. An interesting theory was revealed in 1977 by Cockayne and Hedetniemi, newlinethey made outstanding survey related to dominating set in graphs. The writing regarding newlinethis matter has been surveyed and point by point in the two excellent domination books newlineby Hynes, Hedetniemi and Slater who made a remarkable showing with regards to of newlinebringing together outcomes scattered through somewhere in the range of more than 1200 newlinedomination papers around them. Recent books on domination has stimulated sufficient newlineinspiration leading to the expansive growth of this field of study. newlineA whole host of additional variations of domination can be obtained by imposing newlinerestrictions on the dominating set. Muthammai, Bhanumathi and Vidhya introduced the newlineconcept complementary tree domination. Distance k-dominating set was defined by newlineHenning etal. Another parameter called complementary nil domination was defined by T. newline