  We provide a systematic construction of string junctions in curved
backgrounds which are relevant in computing, within the gauge/gravity
correspondence, the interaction energy of heavy dyons, notably of
quark-monopole pairs, in strongly coupled SYM theories. To isolate the
configurations of physical interest we examine their stability under small
fluctuations and prove several general statements. We present all details, in
several examples, involving non-extremal and multicenter D3-brane backgrounds
as well as the Rindler space. We show that a string junction could be
perturbatively stable even in branches that are not energetically the most
favorable ones. We present a mechanical analog of this phenomenon.
