Number of combinations without given pairs
Don't expect an efficient algorithm.  This is the problem of counting the number of independent sets in an undirected graph.  This problem is #P-complete [1,2], so there is unlikely to be any effective algorithm that works for arbitrary graphs.
See https://cstheory.stackexchange.com/q/10700/5038 for techniques you could use.
[1] The complexity of counting cuts and of computing the probability that a graph is connected, J. Scott Provan and Michael O. Ball, SIAM J. Computing, 12:4.
[2] The complexity of counting in sparse, regular, and planar graphs, Salil P. Vadhan, SIAM J. Computing, 31:2(398--427).