The author considers nine of the fourteen triangle singularities in Arnold’s list [cf. V. T. Arnold, Invent. Math. 35, 87-109 (1976; Zbl 0336.57022)]. Namely the singularities \(E_{12}\), \(E_{13}\), \(E_{14}\), \(Z_{11}\), \(Z_{12}\), \(Z_{13}\), \(Q_{10}\), \(Q_{11}\), \(Q_{12}\). He defines elementary and tie transformations on Dynkin graphs with several components. The aim of this paper is to give a description of Dynkin graphs \(\Gamma\) with several components such that there exists a small deformation fibre \(Y\) of a triangle singularity and satisfying two technical conditions, in terms of elementary and tie transformations of certain Dynkin graphs. The author recalls the definition of a root system of type \(BC\), \(G\) and also explains why this method does not apply to the study of the remaining triangle singularities, namely \(W_{12}\), \(W_{13}\), \(S_{11}\), \(S_{12}\) and \(U_{12}\). He announces that some results on these singularities will appear elsewhere.