Decompositions of graphs
The problem: to decompose V into a partition of components, in each of them there is a path from any vertex to any other vertex.
Depth-first search: procedure dfs (repeated exploring)
Decomposite a undirected graph into connected components (subgraphs): DFS with a counter for each exploring
Previsit and postvisit orderings, example, property, an ancestor interval includes an descendant interval.
Types of edges, determined by interval
Cycle detection by back-edge checking: in a dag, every edge leads to a vertex with a lower post number
Source node and sink node: every dag has at least one of each
Linearlization, topological sorting
The meta-graph of a directed graph is a dag
Run explore within a sink component will visit all nodes in that component and no other nodes
After a DFS starting from any node, the node with the highest post number must be in a source component of the meta-graph. Proof
To turn a source node to a sink node, we can run DFS on the reverse graph of G
After a sink component is identified, it is removed from G, and the process is repeated
