Arborescence of a graph - Mathematics Stack Exchange An arborescence in $G$ rooted at $r$ is a subgraph $H$ of $G$ in which every vertex $u \in V \backslash \ {r\}$ has a directed path to the special vertex $r$ The weight of an arborescence $H$ is the sum of the weights of the edges in $H$
What is an example of a DAG that is *not* an arborescence? An arborescence is a directed graph where there exists a vertex r (called the root) such that, for any other vertex v, there is exactly one directed walk from r to v (noting that the root r is unique)
What is the equivalent of a tree for directed graphs? An arborescence is a tree in which every vertex other than the root has an in-degree of exactly one […] An arborescence is in a sense a tree directed out of the root Therefore an arborescence is sometimes referred to as an out-tree (Reversing the direction of every edge in an arborescence will produce what may be called an in-tree
Number of spanning arborescences does not depend on $i$. An arborescence is a tree with the extra condition that all edges are directed to the root So from $x_i$ there is a unique path to the root, and the first edge of that path is directed out of $x_i$
Graph Theory Algorithm - Playoffs - Mathematics Stack Exchange The question isn’t entirely clear Do you know the results of all possible pairings? And are there any limitations on how you arrange the tree? Are you allowed, for instance, to make it horribly unbalanced, so that the desired winner plays only one match?