yEd supports the notion of "nesting" nodes. I.e. in yEd your white, dark blue, and light green nodes should be group nodes. Moreover, the dark blue node should be a child node of the white node, the light green and light blue nodes should be child nodes of the dark blue node, and the red nodes should be child nodes of the light green node. With such a setup, the edges connecting child nodes will be properly rendered above the parent node.
Please see chapter Creating a Graph in the yEd Manual for information on group nodes and how to assign nodes as child nodes.