Is "Connected Component" unique for each graph? Definition

A connected component of an undirected graph $G$ is a subgraph where any two vertices are connected by paths. A connected component is a maximal connected subgraph in $G$.

Consider a complete graph $K_3$ and its subgraphs. I cannot understand the word maximal so I get the following question that I want to get confirmed by the subquestions to be totally certain about the term connected component.

Is a connected component unique?

  
*
  
*Is the connected component the triangle graph for all different subgraphs of $K_3$ (yes)?
  
*Does there exist other connected components for some subgraphs of $K_3$?
2.1. Are the V-shaped graphs the connected components of the triangle graph at the top (a path exists to connect all vertices)?
2.2. Or is the only connected component the maximal subgraph, the triangle graph here?
  
*What does the word maximal really mean here? Not required to be unique (so not using the term maximum connected component)? The triangle graph is the maximum connected component of all subgraphs of $K_3$?

 A: Your "Definition" underlaid in khaki is slightly lopsided.
Given an undirected graph $G=(V,E)$ call two vertices $x$, $y\in V$ equivalent if there is an edge path connecting $x$ and $y$ in the obvious way. Each equivalence class $C_\iota\subset V$ together with the edges connecting the $x$, $y\in C_\iota$ is called a connected components of $G$. If there is just one such component, i.e., if any two vertices $x$, $y\in V$ can be connected by an edge path in $G$, then the graph $G$ is called connected.
Given a graph $G$ there are many subgraphs $G'$ of $G$, some connected, some consisting of several connected components. It is easy to see that a connected component $C=(V',E')$ of $G$, in the sense defined in the first paragraph, is a maximal connected subgraph of $G$, since there is no vertex $z\in V\setminus V'$ that is connected to a single vertex in $V'$, let alone to all of them.
A: A connected component is not necessarily unique. In the graph, $H$, below:

we have two connected components $G_1$ and $G_2$.
However, most of the graphs you will encounter in a graph theory course are connected and thus only have 1 unique connected component.
Now to answer your question.
1 and 2. The first graph has the triangle graph as its connected component. The second level graphs have the V graph as their connected component. The third level graph has two connected components: a single vertex and a the two vertices connected by an edge. The fourth level graph has three connected components each a single vertex.


*A connected component, $G$ is maximal in the sense that there are no other subgraphs that contain $G$ and are connected. (As per @Fimpellizieri)

A: Yes, for a given graph, the connected components are uniquely defined (there can be several CC per graph).
The definition is given in terms of connected subgraphs, i.e. subgraphs "made of a single piece". A connected subgraph is maximal, hence is a connected component, if you can't make it larger, i.e. if no other vertex is connected to it.
Consequently, every connected subgraph is included in a single connected component.

Q1 and Q2: Among the $5$ graphs, the three on the top have a single CC, the middle one has two and the third one three. (An isolated vertex is a CC.)
Q3. In the top graph, a subgraph made of two vertices isn't maximal as it is connected to the third vertex.
