On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs

Abstract

We study a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) and continued in Simson and Zając (2017) and Makuracki and Simson (2019). In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥1 vertices Δ0={a1,…,an+r} of corank r≥0, that is, bigraphs with the symmetric Gram matrix GΔ≔12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ.One of the main results of this paper asserts that every such a bigraph Δ with n+r≥1 vertices is algorithmically reduced by using at most O(n2) inflation operators tab−, a,b∈Δ0, to one of the canonical r-vertex extensions D̂n(r)∈{Ân(r),D̂n(r),Ê6(r),Ê7(r),Ê8(r)} of the simply laced Dynkin diagram Dn∈{An,Dn,E6,E7,E8}, where Dn is the Dynkin type of Δ, that is, a unique simply laced Dynkin graph Dn=DynΔ∈{An,Dn,E6,E7,E8} associated with Δ. We also construct inflation algorithm that: (a) computes in quadratic time a Dynkin type DynΔ of Δ, and (b) constructs a matrix B∈Mn+r(Z), with detB=±1, defining the weak Gram Z-congruence Δ∼ZBD̂n(r), that is, satisfying the equation GD̂n(r)=Btr⋅GΔ⋅B.