Trivalent Graph isomorphism in polynomial time

4 Implementation test 35 Sumario 41 Bibliography 47 3 Preamble The graph isomorphism problem has a long history in mathematics and computer science, and more recently in fields of chemistry and biology. Graph theory is a branch of mathematics started by Euler as early as 1736 with his paper The seven bridges of Könisberg. It took a hundred years before other important contribution of Kirchhoff had been made for the analysis of electrical networks. Cayley and Sylvester discovered several properties of special types of graphs known as trees. Poincaré defined what is known nowadays as the incidence matrix of a graph. It took another century before the first book was published by Dénes K˝ onig at 1936 titled Theorie der endlichen und unendlichen Graphen. After the second world war, further books appeared on graph theory, for example the books of Ore, Behzad and Chartrand, Tutte, Berge, Harary, Gould, and West among many others. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. Besides it's practical importance, the graph isomorphism problem it's one of few problems which belonging to NP neither known to be solvable in polynomial time nor NP-complete. It is one of only 12 such problems listed by Garey & Johnson(1979), and one of only two of that list whose complexity remains unresolved (the other being integer factorization). It is known this computational problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. Since the graph isomorphism problem is neither known to be NP-complete nor to be tractable, researchers have sought to gain insight into the problem by defining a new class GI, the set of problems with a polynomial-time Turing reduction to the graph isomorphism problem [5]. In fact, if the graph isomorphism problem is solvable in polynomial time, then GI would equal P. The best current theoretical algorithm is due to Eugene Luks (1983) and is based on the earlier work by Luks (1981), Babai and Luks (1982), combined with a subfactorial algorithm due to Zemlyachenko (1982). The algorithm relies on the classification of finite simple groups, without these results a slightly weaker bound 2 O(√ n log 2 n) was obtained first for strongly regular graphs by László Babai (1980), and then extended to general graphs by Babai and Luks (1982), where n …