Algorithms and computational complexity of clique and biclique covers I've been reading a paper by a mathematical chemist.  He proposes some indices to measure the complexity of molecules.  From here on in, instead of molecules, think undirected connected graphs: a vertex is an atom, and an edge is a bond between atoms.  It's possible to consider colorings of the vertices, to tell the Nitrogens apart from the Carbons, say, but I'll ignore that part for now.

The point: the indices he suggests are motivated by heuristics and experimental "looks good so far"-ness.  I think there must be some actual theory known about some of these quantities, and I'm hoping to get some pointers here.

Fix a graph $G$.  Let $C$ and $C'$ be two covers of $G$.  Say $C$ and $C'$ are the same kind of cover if they contain the same types of subgraphs in equal numbers.  (Note $C$ and $C'$ do not have to be isomorphic.)  Now we define the following quantities:

$k_S(G) =$ number of kinds of minimal edge clique covers of $G$
$k_T(G) =$ total number of minimal edge clique covers of $G$
$k^{bi}_S(G) =$ same as $k_S(G)$ but for bicliques
$k^{bi}_T(G) = $ same as $k_T(G)$ but for bicliques
$p_S(G) =$ number of kinds of partitions of the edges of $G$ into cliques
$p_T(G) =$ total number of partitions of the edges of the graph into cliques
$p^{bi}_S(G), p^{bi}_T(G)$ as above, but with partitions of $G$ into bicliques

Empirically, it is apparently easier to calculate the $p$ measures than it is to calculate the $k$ measures.  I expect something must be known somewhere about calculating some of these quantities.  Can anyone provide algorithms, computational hardness, etc.?  Thanks.
I am not sure if this is what you need.

Both Edge Clique cover and Edge Clique partition are in FPT, see

Data reduction and exact algorithms for clique cover

Parameterized complexity of the clique partition problem

Biclique cover is also studied and shown to be in FPT.