  We introduce the continuous-time vertex-reinforced random walk (cVRRW) as a
continuous-time version of the vertex-reinforced random walk (VRRW), which
might open a new perspective on the study of the VRRW.
  It has been proved by Limic and Volkov that for the VRRW on a complete-like
graph $K_d \cup \partial K_d$, the asymptotic frequency of visits is uniform
over the non-leaf vertices. We give short proofs of those results by
establishing a stochastic approximation result for the cVRRW on complete-like
graphs. We also prove that almost surely, the number of visits to each leaf up
to time n divided by $n^{\frac{1}{d-1}}$ converges to a non-zero limit. We
solve a conjecture by Limic and Volkov on the rate of convergence in the case
of the complete graph.
