THE CLASSIFICATION PROBLEM FOR GRAPHS AND LATTICES IS WILD RUVIM LIPYANSKI AND NATALIA VANETIK Abstract. We prove that the classification problem for graphs and several types of algebraic lattices (distributive, congruence and modular) up to isomorphism contains the classification problem for pairs of matrices up to simultaneous similarity. 1. Introduction The graph isomorphism problem is one of the central problems in graph theory. Reduc- ing this problem to the isomorphism problem for algebraic structures, such as rings, al- gebras and groups, was studied in several works, e.g., [Kim, Roush ’80], [Droms ’87] and [Saxena, Agrawal ’05]. The classification problem for finite and infinite algebraic lattices has also been extensively addressed. A wild classification problem contains the problem of classification of pairs of matrices up to simultaneous similarity. In this paper, we prove that the classification problem for graphs is wild by reducing the classification problem for finite 2-nilpotent p-groups to the classification problem for graphs (the wildness of classification problem for finite 2-nilpotent p-groups was proved in [Sergeichuk ’75]). We use wildness of the classification problem for graphs to show that the classification problem for algebraic lat- tices and poset lattices is wild. A reduction from graphs to lattices, described in this paper, allows us to prove that the classification problem for distributive, modular and congruence lattices is wild, even for finite lattices. 2. Wildness We use in this paper the following definitions of a matrix problem and wildness, first given in [Belitskii, Sergeichuk ’03]. A matrix problem {A 1 , A 2 } is a pair that consists of a set A 1 of a-tuples of matrices from M n×m , and a set A 2 of admissible matrix transformations. Given two matrix problems A =(A 1 , A 2 ) and B =(B 1 , B 2 ), we say that the matrix problem A is contained in the matrix problem B if there exists a b-tuple T (x)= T (x 1 ,...,x a ) of matrices, whose entries are non-commutative polynomials in x 1 ,...,x a , such that (1) T (A)= T (A 1 ,...,A a ) ∈B 1 if A =(A 1 ,...,A a ) ∈A 1 ; 1991 Mathematics Subject Classification. 05C60, 05C25, 7B30, 06B20, 06D50, 06C05, 15A21. Key words and phrases. wild problems, graphs, lattices. 1