To review, a **directed** graph consists of edges that can only be traversed in one direction. Additionally, a **acyclic** graph defines a graph which does not contain cycles, meaning you are unable to traverse across one or more edges and return to the node you started on. Putting these definitions together, a **directed acyclic** graph, sometimes abbreviated as DAG, is a graph which has edges which can only be traversed in one direction and does not contain cycles.
A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a directed acyclic graph is a linear ordering of its vertices such that for every directed edge $u\to v$ from vertex $u$ to vertex $v$, $u$ comes before $v$ in the ordering.
The BFS version, known as [Kahn's Algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm), makes it obvious how to extract the lexicographically minimum topological sort.
<resource source="PAPS" title="9.1">Best Path in a DAG</resource>
</resources>
One useful property of directed acyclic graphs is, as the name suggests, that no cycles exist. If we consider each node in the graph as a state, we can perform dynamic programming on the graph if we process the states in an order that guarantees for every edge $u\to v$ that $u$ is processed before $v$. Fortunately, this is the exact definition of a topological sort!
Let $dp[v]$ denote the length of the longest path ending at the node $v$. Clearly
$$
dp[v]=\max_{\text{edge } u\to v \text{ exists}}dp[u]+1,
$$
or zero if $v$ has in-degree $0$. If we process the states in topological order, it is guarangeed that $dp[u]$ will already have been computed before computing $dp[v]$.
<!-- However, not all problems clearly give you directed acyclic graphs (ex. [Plat - Cave Paintings](http://usaco.org/index.php?page=viewproblem2&cpid=996)). An important step in many problems is to reduce the statement into a directed acyclic graph. See the editorial of the linked problem for more information.
