Single- and double-stranded DNA are increasingly being paired with surfaces and nanoparticles for numerous applications, such as sensing, imaging, and drug delivery. Unlike the majority of DNA structures in bulk that are stabilized by canonical Watson-Crick pairing between Ade-Thy and Gua-Cyt, those adsorbed on surfaces are often stabilized by noncanonical base pairing, quartet formation, and base-surface stacking. Not much is known about these kinds of interactions. To build an understanding of the role of non-Watson-Crick pairing on DNA behavior near surfaces, one requires basic information on DNA base pair stacking and hydrogen-bonding interactions. All-atom molecular simulations of DNA bases in two cases-in bulk water and strongly adsorbed on a graphite surface-are conducted to study the relative strengths of stacking and hydrogen bond interactions for each of the 10 possible combinations of base pairs. The key information obtained from these simulations is the free energy as a function of distance between two bases in a pair. We find that stacking interactions exert the dominant influence on the stability of DNA base pairs in bulk water as expected. The strength of stability for these stacking interactions is found to decrease in the order Gua-Gua > Ade-Gua > Ade-Ade > Gua-Thy > Gua-Cyt > Ade-Thy > Ade-Cyt > Thy-Thy > Cyt-Thy > Cyt-Cyt. On the other hand, mutual interactions of surface-adsorbed base pairs are stabilized mostly by hydrogen-bonding interactions in the order Gua-Cyt > Ade-Gua > Ade-Thy > Ade-Ade > Cyt-Thy > Gua-Gua > Cyt-Cyt > Ade-Cyt > Thy-Thy > Gua-Thy. Interestingly, several non-Watson-Crick base pairings, which are commonly ignored, have similar stabilization free energies due to interbase hydrogen bonding as Watson-Crick pairs. This clearly highlights the importance of non-Watson-Crick base pairing in the development of secondary structures of oligonucleotides near surfaces.