A present trend in the study of the Symmetric Traveling Salesman Polytope (STSP(n)) is to use, as a relaxation of the polytope the graphical relaxation (GTSP(n)) rather than the traditional monotone relaxation which seems to have attained its limits. In this paper, the authors show the very close relationship between STSP(n) and GTSP(n). In particular, they prove that every non-trivial facet of STSP(n) is the intersection of n+1 facets of GTSP(n), n of which are defined by the degree inequalities. This fact permits the authors to define a standard form for the facet-defining inequalities for STSP(n), that they call tight triangular, and to devise a proof technique that can be used to show that many known facet-defining inequalities for GTSP(n) define also facets of STSP(n). In addition, the authors give conditions that permit to obtain facet-defining inequalities by composition of facet-defining inequalities for STSP(n) and general lifting theorems to derive facet-defining inequalities for STSP(n+k) from inequalities defining facets of STSP(n).
