Graphs encoding 3-manifolds of genus 2

We give a simple alternative proof of the representation theorem of all genus two 3-manifolds by a 6-parameter family of integers, due to Casali and Grasselli (see [M.R. Casali, A catalogue of the genus two 3-manifolds, Atti. Sem. Mat. Fis. Univ. Modena 37 (1989) 207–236]; [M.R. Casali, L. Grasselli, 2-Symmetric crystallizations and 2-fold branched coverings of S3S3, Discrete Math. 87 (1991) 9–22]). Our approach is different from that of the quoted papers, and it is based on the concept of extended Heegaard diagram. This permits to obtain new topological meanings of the arithmetic conditions which the parameters must satisfy for representing closed manifolds. Some applications and results on some homology spheres of genus 2 complete the paper.