A construction of chromatic index critical graphs

We prove that if K is an undirected, simple, connected graph of even order which is of class one, regular of degree p ≥ 2 and such that the subgraph induced by any three vertices is either connected or null, then any graph G obtained from K by splitting any vertex is p-critical. We find that various constructions of critical graphs by S. Fiorini are special cases of a corollary of this result.