English

 Abstract: In [R-S], Rubinstein-Scharlemann introduced a powerful machinery , what is called {\it graphic}, for studying Heegaard splittings of 3-manifolds, and obtained deep results on Reidemeister-Singer distance of two strongly irreducible Heegaard splittings of a 3-manifold. I will give a introductory talk on this concept, then give some explanations on some topics which are related to them (Bachman's work on critical Heegaard splittings, recent work by Rieck-Rubinstein on the Heegaard genuses of 2-fold branched covering spaces of 3-maniolds etc.) [R-S] H. Rubinstein and M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35(1996), 1005--1026.