For any connected graph $G=(V,E)$, we must have $|E|\geq|V|-1$. Any edges less, and the graph wouldn't be connected. It's sorta like saying a tree graph is a "local minimum" in the space of graphs, where height is determined by the edge count. This is akin to minimal surfaces, where such a surface is a local minimum in the variation of family of regular surfaces. Is there a connection between the two?
2026-03-26 14:19:55.1774534795
Is there a connection between minimal surfaces and tree graphs?
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Kind of. For this analogy, the comparison function between graphs would be the inclusion ; and the constraints would be connectivity-related properties.
There are lots of constraints you can have for different problems. If you want your tree to be below a particular graph, then your local minima below this are the spanning trees associated with that graph. If each edge has a cost, you can also find a global minimum (the MST, minimum spanning tree) with algorithms such as Kruskal's or Prim's. If your constraint is just that a specified subset $S \in V$ of vertices is in the same connected component, then you're looking for a Steiner tree (this is a famous NP-complete problem).