Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal solution for the problem on $G$, then $S \wedge G_i$ is a solution on $G_i$ all $G_i \subseteq G$. Is there a canonical name for this property for a graph optimization problem?
A master solution for an instance of a combinatorial problem is a solution with the property that it is optimal for any sub instance.
Martijn van Ee, René Sitters:
On the Complexity of Master Problems.
Proceedings of MFCS-2015, LNCS 9235, pp 567-576
Deineko, V.G., Rudolf, R., Woeginger, G.J.:
Sometimes travelling is easy: The master tour problem.
SIAM Journal on Discrete Mathematics 11, 1998, pp 81–93.