Prove Lemma 1

Definition 1 Let g = (V. E) be a directed graph, and let rooted, directed, spanning tree (RDST) o g is a subgraph T hold: V be a distinguished node that we call the root. A (V. E ) o g (that is. E, C E) such that two conditions 1. Ifwe ignore the directions of the edges in F', then T is a spanning tree of g; and, 2. If we take the directions of the edges in E into account, then there is a (directed) path from the root r to every node v E V Definition 2 Let g (V, E, w) be a weighted, directed graph with weight function w:ER+ (here R+ is the set of all non-negative reals) and root node r. A rooted, directed minimum spanning tree, or RDMST, is an RDST of g that has the smallest weight among all possible RDST's of g. Lemma 1 Let g -(V.E,w) be a weighted, directed graph with designated root r e V. Let E' = {me(u) : u E (V \ {r))). Then, either T = (V, E') is an RDMST of g rooted at r or T contains a cycle. Definition 1 Let g = (V. E) be a directed graph, and let rooted, directed, spanning tree (RDST) o g is a subgraph T hold: V be a distinguished node that we call the root. A (V. E ) o g (that is. E, C E) such that two conditions 1. Ifwe ignore the directions of the edges in F', then T is a spanning tree of g; and, 2. If we take the directions of the edges in E into account, then there is a (directed) path from the root r to every node v E V Definition 2 Let g (V, E, w) be a weighted, directed graph with weight function w:ER+ (here R+ is the set of all non-negative reals) and root node r. A rooted, directed minimum spanning tree, or RDMST, is an RDST of g that has the smallest weight among all possible RDST's of g. Lemma 1 Let g -(V.E,w) be a weighted, directed graph with designated root r e V. Let E' = {me(u) : u E (V \ {r))). Then, either T = (V, E') is an RDMST of g rooted at r or T contains a cycle