Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition I. Inflation Algorithm Revisited *

Abstract

We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph Δ, or equivalently, on the associated integral quadratic form q_Δ : ℤ ⁿ → ℤ, preserving Gram ℤ-congruence. The ideas are inspired by classical results of Ovsienko and recent studies of Simson started in [SIAM J. Discr. Math. 27 (2013), 827-854], concerning classifications of integral quadratic and bilinear forms, and their Coxeter spectral analysis. We provide few modifications of the inflation algorithm and new estimations of its complexity for positive and principal loop-free bigraphs. We discuss in a systematic way the behavior and computational aspects of inflation techniques. As one of the consequences we obtain relatively simple proofs of several interesting properties of quadratic forms and their roots, extending known facts. On the other hand, the results are a first step of a solution of a variant of Grothendieck group recognition, a difficult combinatorial problem arising in representation theory of finite dimensional algebras and their derived categories, which we discuss in Part II of this two parts article with the same main title.