A weighted graph is star-convex precisely when it contains a star-convex tree spanning all leaves, and convex sequences embed into spider graphs to produce star-convexity.
Minimal (max,+) realization of convex sequences
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On Weighted Star--Convex Graphs
A weighted graph is star-convex precisely when it contains a star-convex tree spanning all leaves, and convex sequences embed into spider graphs to produce star-convexity.