The block treedepth of a graph 
is the minimum number of rounds of vertex
deletions needed to eliminate all edges when blocks
are treated independently. In each round, replace each remaining graph
by separate copies of its blocks, then delete one
vertex from each copy containing an edge
(Giannopoulou and Mavropoulos 2024).
This process gives a recursive computation with base value 0 for an empty graph, which requires no deletions. For any other graph, the block treedepth is the maximum of the block treedepths of its blocks if there is more than one graph block, or one plus the minimum block treedepth obtainable by deleting a vertex if there is only one graph block.
This is the same recursive construction as 2-treedepth, with a different base value. Comparing the definitions gives
 |
for every graph 
with at least one vertex.
Both parameters are 0 for the 0-vertex graph. Thus block
treedepth and 2-treedepth are equivalent up to this
normalization, but their numerical values should not be identified without checking
the convention (Huynh et al. 2022, Giannopoulou and Mavropoulos 2024).
