  S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an
inclusion based on induced subgraphs. This conjecture has now been settled
completely by M. Chudnovsky and P. Seymour. One part of the proof proves the
result for the bounded case, a result proved independently by C. J. Altomare.
We give two short proofs of the bounded case of S. B. Rao's conjecture. Both
the proofs use the fact that if the number of entries in an integer sequence
(with even sum) is much larger than its highest term, then it is necessarily
graphic.
