Abstract
A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN groups and free products with amalgamation. The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical. For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem. These results generalise the known ones for HNN groups and free products with amalgamation.
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