One of the main aims of the paper is to develop the mesh geometry technique for corank-two edge-bipartite graphs Î with n+2â„3 vertices, and the mesh algorithms introduced in [30,33] and successfully studied in our recent article [42]. We introduce and study the concept of a self-duality of mesh geometries Î(RËÎ,ΊÎ) viewed as ΊÎ-mesh translation quivers. We show how self-dualities of mesh geometries Î(RËÎ,ΊÎ) and the mesh geometry technique is applied to an affirmative algorithmic solution of so called Horn-Sergeichuk type problem [9, Problem 4.3] on the self-congruency of square integer matrices AâMn+2(Z), for the class of non-symmetric Gram matrices A=GËÎ of corank-two loop-free edge-bipartite graphs Î, with n+2â€6 vertices. More precisely, we show that each of the mesh geometries Î(RËÎ,ΊÎ) is self-dual, we construct its dual form Î(RËÎ,ΊÎ)op=Î(RËÎ,ΊÎâ1) isomorphic with Î(RËÎ,ΊÎ), and we construct a canonical self-duality isomorphism fÎ:Î(RËÎ,ΊÎ)âÎ(RËÎ,ΊÎ)op of mesh translation quivers. Using the self-duality fÎ we construct combinatorial algorithms such that, given a square Gram matrix A=GËÎâMn+2(Z) of Î lying in this class, they are able to compute a Z-invertible matrix BâMn+2(Z) that coincide with its inverse Bâ1 and defines the congruence of A with Atr, i.e., the equation Atr=Btrâ
Aâ
B is satisfied.An idea of our solution is outlined in Section 8 of our recent article [42], where among others two of our 13 algorithms solving the problem are constructed. The remaining 11 algorithms are constructed in the present article. We do it by means of the structure of the standard self-dual ΊÎ-mesh translation quiver Î(RËÎ,ΊÎ) (called a geometry) canonically associated with Î, consisting of ΊÎ-meshes of ΊÎ-orbits OÎ(w) of vectors wâRËÎâZn+2, where ΊÎ:Zn+2âZn+2 is the Coxeter transformation of Î. We construct in the paper such self-dual ΊÎ-mesh geometry Î(RËÎ,ΊÎ), for each of the corank-two loop-free edge-bipartite graphs Î, with n+2â€6 vertices.