  The graph parameter vertex integrity measures how vulnerable a graph is to a
removal of a small number of vertices. More precisely, a graph with small
vertex integrity admits a small number of vertex removals to make the remaining
connected components small. In this paper, we initiate a systematic study of
structural parameterizations of the problem of computing the
unweighted/weighted vertex integrity. As structural graph parameters, we
consider well-known parameters such as clique-width, treewidth, pathwidth,
treedepth, modular-width, neighborhood diversity, twin cover number, and
cluster vertex deletion number. We show several positive and negative results
and present sharp complexity contrasts.
