[Math] Definition of component for a digraph

Question

I could find this in Wikipedia

Component: A connected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have some higher order of connectivity, including biconnected components, triconnected components, and strongly connected components.

which I haven't yet understoond in a directed graph. I am studying the Handbook of Graph Theory by Gross et all. You may use the following example for demonstrations. So

What is the definition of component for a digraph?

Answer 1

A connected component is usually only defined on undirected graphs. (see below). For the more general digraphs we use the concept of strongly connected components (see below) which generalize the notion of connected components to directed graphs. Sometimes we also talk about weak connectedness which is just the (undirected) connectedness if when we consider the graph where we replace the directed edges with undirected ones.

In graph theory, a connected component (or just component) of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph.

This means the subgraph we are talking about does have to meet following criterion:

• For any two vertices in this subgraph there is a path that begins with one and ends in the other.

The second part is merely a corollary.

https://en.wikipedia.org/wiki/Connected_component_%28graph_theory%29

For digraphs we talk about strongly connected components:

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them.

https://en.wikipedia.org/wiki/Strongly_connected_component