The 2-treedepth 
of a graph 
is the block-based analog of treedepth.
It is 0 for the 0-vertex graph, the maximum value over
the blocks when there is more than one graph
block, and
 |
when 
consists of a single graph
block. Here isolated vertices and graph
bridges with their endpoints also count as blocks
(Huynh et al. 2022, Hodor et al. 2026).
Huynh et al. (2022) introduced the parameter with notation 
using a vertex coloring
characterization. Hodor et al. (2026) use the name "2-treedepth."
The related term block treedepth uses a different
normalization in Giannopoulou and Mavropoulos (2024). For a graph
with at least one vertex, their block
treedepth is one less than the 2-treedepth defined here.
An empty graph on at least one vertex has 2-treedepth 1, every forest containing an edge
has 2-treedepth 2, and the complete graph 
has 2-treedepth 
. The 2-treedepth never exceeds the treedepth,
but the path graphs have unbounded treedepth
and 2-treedepth at most 2.
Hodor et al. (2026) prove equality of the two parameters for graphs with no induced four-vertex path graph. More generally, excluding an induced path graph of fixed vertex count bounds the treedepth by a polynomial in the 2-treedepth.
