  We consider edge modification problems towards block and strictly chordal
graphs, where one is given an undirected graph $G = (V,E)$ and an integer $k
\in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to
obtain a block graph or a strictly chordal graph. The completion and deletion
variants of these problems are defined similarly by only allowing edge
additions for the former and only edge deletions for the latter. Block graphs
are a well-studied class of graphs and admit several characterizations, e.g.
they are diamond-free chordal graphs. Strictly chordal graphs, also referred to
as block duplicate graphs, are a natural generalization of block graphs where
one can add true twins of cut-vertices. Strictly chordal graphs are exactly
dart and gem-free chordal graphs. We prove the NP-completeness for most
variants of these problems and provide $O(k^2)$ vertex-kernels for Block Graph
Edition and Block Graph Deletion, $O(k^3)$ vertex-kernels for Strictly Chordal
Completion and Strictly Chordal Deletion and a $O(k^4)$ vertex-kernel for
Strictly Chordal Edition.
