I have a problem that consists of finding a good solution for a graph that has vertex weights (and this cost is the highest priority), but also has edge costs.
One solution would be to convert the original graph to a graph with only edge costs, and then use Dijkstra's algorithm.
To do this we can replace a single node with n nodes which are fully connected to each other. n is the number of edges entering that node.
Note: This solution assumes that weights are non negative.