Is This Perfect Forward Secrecy?

Is This Perfect Forward Secrecy?

I have a textbook that contains the following problem:

In practice, one master key, K_M, is exchanged in a secure way (e.g. Diffie-Hellman key exchange) between the involved parties. Afterwards, session keys, k_x, are regularly updated by using key derivation. Consider the following three different methods of key derivation:

a. k₀ = K_M; k_i+1 = k_i+1

b. k₀ = H(K_M); k_i+1 = H(k_i)

c. k₀ = H(K_M); k_i+1 = h(K_M || i || k_i)

where H() represents a secure hash function, and k_i is the i^th session key.

The question goes on to ask which method(s) provide Perfect Forward Secrecy (PFS), with the answer key stating that b and c do.

As I understand it, a requirement of PFS is that past sessions are secure even if the long-term secret is compromised [1]. However, in this case, if K_M (the long-term secret) is compromised, then all session keys can be computed.

Do any of these key derivation schemes provide PFS? Does the particular wording of the problem, ...exchanged in a secure way..., remove my assumption that K_M could be compromised? Is K_M not a long-term secret in the context of PFS?