An Encounter with Graphs

So far diagnosis of some problems we came across during our works with algorithms, graphs played significant roles. In this paper, we have tried to show that graphs really occupy a major role in computer science and engineering. The abstraction of problems as different graph models as well as graphs in digital system design methodology allows us to have a close look or refresher outlook on graphs. We have also touched root level, historical purview, graphs in contemporary Indian scenario, and a significant aspect of perfect graphs in VLSI area. We observe that hard computing does not suit in most of graph problems. Soft computing approaches are often a tool in graph algorithm design. A pessimal algorithm also came across our route that is really an intellectual property to share with the readers. To part, with we remind that no diagnosis is final and that pathology is in fact always eager to have a new look. 1. Prologue [1, 2, 5, 6, 11, 13, 21, 23, 24] This paper is an honest, documented cursory glance to application and interest in graph theory for last about 300 years. This is merely an introduction and invitation. We hope the taste will linger after presentation of the text and certain audience may even take the help of the references for further studies. We will be rewarded if we have inspired somebody a little. We announce the prologue and retire for the time being, after us better players will arrive. 2. In Lieu of an Introduction [1, 2, 5, 6, 11, 13, 21, 23, 24] A backtrack to the History of Graphs: Graph theoretic folklores are so widely publicized that natural human inclination will insist on seeing evolution of graph algorithms in reference to these folklores. Hence we have an excuse (not lame in any way) for the backtracking. Journal of Physical Sciences, Vol. 10, 2006, 188 – 200 An Encounter with Graphs 189 The four fathers of graph theory Euler, Kirchhoff, Cayley and Hamilton (in chronological order) discovered graph theory while trying to solve either puzzle or a problem of the physical world. The graph theory was christened as a mathematical discipline in 1736 with the first paper of Euler on this subject. He solved (or rather proved it unsolvable) the Konigsberg Bridge Problem by translating it in terms of graphs. Next 100 years were almost a blank period for graph theory. In 1847, Kirchhoff developed the theory of trees in order to solve the electrical network equations. Although a physicist, he ably abstracted electrical networks like mathematician in terms of graphs. Ten years later the number of isomers of saturated hydrocarbons (CnH2n+2) was expressed by Arthur Cayley as the graphical problem of enumerating the class of trees in which each point is of degree 1 or 4. An elegant and general solution was first obtained by Polya, who developed powerful enumeration technique. Extremely effective use of Polya’s theorem has been made by the theoretical physicists R. W. Ford and G. E. Uhlenbech in (1956) solving several enumeration problems for graphs which arises in a study of statistical mechanics. A game invented by Sir William Hamilton in 1859 uses a regular dodecahedron whose 20 vertices are labeled with the name of famous cities. The player is challenged to travel around the world by finding a close circuit along the edges that passes through each vertex exactly once. In fact, in terms of graphs it demanded the necessary and sufficient condition for the existence of Hamiltonian circuits in an arbitrary graph. The most famous unsolved problem in graph theory is the celebrated Four Color Conjecture. It states that four colors are sufficient to color any atlas (a map on a plane) such that no countries with common boundaries have same colors. The conjecture has an interesting history. In a letter to Sir W. R. Hamilton dated 23rd October 1852, Professor Augustus de Morgan mentioned the problem and also printed out the name of his student Cutwise as its originator. The conjecture appeared in print for the first time in June 1878, in the form of a question posed by the well-known mathematician Cayley. Cayley again stated the problem in the first volume of proceedings of the Royal Geographical Society in 1879. The most recent work on this problem was done by Hakim and Apple of Illinois, USA, in 1976. In 1920, Graph theory arose to a new height due to Konig and others. The real formalization of graph theory was due to Veblen’s work. In 1932, Faster established the connection between geometric networks and electric circuits. It was followed by Whitney’s fundamental work on graph theory. The pace of research in graph theory has accelerated even further in the past two decades. One of the manifestations of this activity was the immediate success of the Journal of Combinatorial Theory founded in 1966. Perhaps the most striking evidence of this upsurge of interest was the first book ever written in graph theory by Konig (1936). No books were published within 1968. Several books and thousands of papers have been published in graph theory in recent years. 190 Rajat Kumar Pal et al. z y x N A N D N A N D