This problem is an analysis of a single$-$player puzzle game played on an undirected graph $$

$$

G$$ called Water$-$Seeker. Each node of $$

$$

G$$ either contains a pure, untapped water source or is empty. The player must open or mark each node as a water source. If the player opens an empty node, the player learns the number of neighboring nodes that contain water. If the player opens a water source node it is contaminated. The player wins if and when all the sources have been identified without contaminating any of the water.

In the puzzle$-$building problem, you are given a graph $$G$$ along with numbers labeling some of its nodes. You must determine whether $$G$$ can be expanded into a consistent Water$-$Seeker game board. In other words, can the remaining nodes of $$G$$ be labeled so that the result is a valid, playable game?