I don't remember right now how to do this, but this is definitely a standard topic. It is most likely proved rigorously in Lewis and Papadimitriou and Sipser. There are also probably places on the Internet that explain this. Are you referring to such places saying that "the answers haven't come up properly"? If you have more specific questions about the proof, we can discuss them.a) How do you show that the Hamiltonian Circuit problem is NP-Complete
Reducing DHC to HC may be somewhat nontrivial (I think I remember seeing a proof), but again I would search the Internet.
Could you remind how graphs are represented as inputs? I believe the input size is linear w.r.t. the number of vertices and edges; is this so?2) An independent set I is a subset of the nodes of a graph G where: no 2 nodes in I are adjacent in G. For natural number k, the problem k-IND asks if there is an independent set of size k. The problem: How do you show k-IND is in Logspace.
Basically, to decide k-IND it is enough to store k identifiers of vertices. One also needs an algorithm that searches through all k-subsets of vertices (i.e., writes each subset, one after another, into the space reserved for k vertices). Another subroutine that is needed is to check, for given k identifiers of vertices, if they are independent. If vertices are numbered in binary, then the space needed to store k identifiers is logarithmic w.r.t. input -- this is the most important part to check.