Suppose I give a solver (CPLEX, Gurobi, SCIP or anything else) an IP which is a reformulation of a stable set problem (or vertex cover problem or coloring problem) of some graph, is there a way I can tell the solver that it is a stable set or vertex cover instance? Will that enhance the heuristics used by the solver?
I suspect there are a few specific problems for which the answer is "yes," and I hope others will answer to provide examples of those.
But in general I believe the answer is "no." For example, if you formulate the minimum-spanning tree problem as an IP and try to solve it with a general-purpose solver, it will be much slower than just using Prim's or Kruskal's algorithm. If there were some option you could set that says "hey, this is an MST!", then the solver would basically have to have a ton of separate graph algorithms (Prim's for MST, Dijkstra's for shortest path, etc.) built into it, which is not really what general-purpose solvers are designed to do.
CPLEX has a parameter (RootAlgorithm) that lets you select the method for solving an LP (or for solving the root node relaxation of an ILP). The default setting is to let CPLEX choose, which usually (but not always) results in it using dual simplex. One of the choices is "network simplex", which you might try for a graph problem. I don't know whether CPLEX would detect the graph structure and automatically try network simplex if left on the default setting.
Often such problems have side constraints, and this patent covers that more general case, using Dantzig-Wolfe decomposition with the network subproblem (MST, TSP, etc.) expressed compactly (not algebraically) and solved with a specialized solver. This functionality is implemented in SAS but currently undocumented. Please contact me if you are interested in using it.