A graph
G is vertex pancyclic if for each vertex
\({v \in V(G)}\) , and for each integer
k with 3 ≤
k ≤ |
V(
G)|,
G has a
k-cycle
C k such that
\({v \in V(C_k)}\) . Let
s ≥ 0 be an integer. If the removal of at most
s vertices in
G results in a vertex pancyclic graph, we say
G is an
s-vertex pancyclic graph. Let
G be a simple connected graph that is not a path, cycle or
K 1,3. Let
l(
G) = max{
m :
G has a divalent path of length
m that is not both of length 2 and in a
K 3}, where a divalent path in
G is a path whose interval vertices have degree two in
G. The
s-vertex pancyclic index of
G, written
vp s (
G), is the least nonnegative integer
m such that
L m (
G) is
s-vertex pancyclic. We show that for a given integer
s ≥ 0,
$vp_s(G)\le \left\{\begin{array}{l@{\quad}l}\qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$
And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.