My "ex cathedra" answer is as follows:
The class P and NP are defined for any chosen and fixed alphabet
set \Sigma and (a particular variant of the) TM model.
It does not concern itself with "interpretations" including how to "encode"
objects such as graphs, trees, lists of numbers, ... into strings over \Sigma.
Such matters are concerned with applications. Whoever invokes
the theory for a particular problem, say in graph theory,
bears the responsibility that encoding such an object to a string in
\Sigma is "reasonable". This of course includes that it being information
theoretically succinct (no padding), and computationally reasonable (I will
interpret as computable and invertible in P-time.) But the official
dogma does not get tangled with this. (I say tangled with... because ,
then the question can be what do you mean by computable and
invertible in P-time.... here we are defining P-time...)
This is similar to the situation in axiomatic set theory.
How to use the language
of set theory to deal with "actual" mathematical problems is
the responsibility of the one uses this language of set theory.
This is how axiomatic set theory "resolves" paradoxes such
as "the set of all sets". It simply does not speak of such things (by
providing no axiom to legitimatize their status as sets), and whoever
invokes such blasphemy bears all the responsibility of inconsistency
(damnation?)
ð
Jin-Yi