# Forest of maximal weight edges in weighted graphs

Consider a weighted undirected graph $$G=(V,E,\omega)$$ and assume for simplicity that all edges have distinct weights.

For all $$v$$ in $$V$$, we denote by $$n(v)$$ the unique neighbor of $$v$$ such that $$\omega(v,n(v)) = \max_{w\in N(v)} \omega(v,w)$$.

Then, the subgraph $$F=(V,E')$$, in which $$E'$$ is the set of edges $$(v,n(v))$$ for all $$v$$, is a forest (a set of trees).

Question: what is the name of this object, and where does it appear in the literature?

Variants appear for instance in:

However, it is not viewed as a fundamental weighted-graph concept in these papers. Are there any other papers dedicated to it? For instance, are there known properties of graph $$G$$ (its structure or its weights) that induce some properties on forest $$F$$ (like its number of trees or their depths)?

• This is Nearest neighbor graph (min instead of max), but not sure there are studies of graph-theoretic properties in general graphs. Commented Jul 9 at 19:26

It could be called the nearest neighbor graph for $$-\omega$$. But I'm afraid that doesn't really tell you all that much...