  In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, where
$K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical
invariants via counting of non-commutative curves. Roughly, these invariants
are sets of subcategories in a given category and their quotients. The
non-commutative curve-counting invariants are obtained by restricting the
subcategories to be equivalent to $D^b(K(l))$. The general definition defines
much larger class of invariants and many of them behave properly with respect
to fully faithful functors. Here, after recalling the definition, we focus on
examples and extend our studies beyond counting. We enrich our invariants with
structures: the inclusion of subcategories makes them partially ordered sets,
and considering semi-orthogonal pairs of subcategories as edges amount to
directed graphs. In addition to computing the non-commutative curve-counting
invariants in $D^b(Q)$ for two affine quivers, for $ A_n$ and $D_4$ we derive
formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$,
whereas for the two affine quivers and for $D_4$ we determine and count all
generated by an exceptional collection subcategories. Estimating the numbers
counting non-commutative curves in $D^b({\mathbb P}^2)$ modulo group action we
prove finiteness and that an exact determining of these numbers leads to
proving (or disproving) of Markov conjecture. Regarding the mentioned structure
of a partially ordered set we initiate intersection theory of non-commutative
curves. Via the structure of a directed graph we build an analogue to the
classical curve complex used in Teichmueller and Thurston theory. The paper
contains many pictures of graphs and presents an approach to Markov Conjecture
via counting of subgraphs in a graph associated with $D^b(P^2)$. Some of the
results proved here were announced in the previous work.
