I am trying to prove the following:
Let $x_1$ be any vertex of a weighted connected graph $G$ with $n$ vertices and let $T_1$ be the subgraph with the one vertex $v_1$ and no edges. After a tree (subgraph) $T_k$, $k
My plan is as follows: Fix an order $\triangleleft$ of the edges of $G$, and order the edges according to the lexicographical order induced by $\triangleleft$ and the cost order; that is, $e_1\lt e_2$ if either $c(e_1)\lt c(e_2)$ or $c(e_1)=c(e_2) \land e_1\triangleleft e_2$. Then show that $T_n$ has the property that each edge $e$ of $T_n$ has ends in different components of $G_e$ where $G_e$ is the graph with the vertices of $G$ and the edges $\lt e$. Then to show that any spanning tree with this property is minimal.
What I cant show is that $T_n$ has this property.
Any help will be appreciated.
Thanks