I am trying to find a Tutte polynomial for a graph. And i am confused with the definition of that as well as the definition of connected component.
From the definition above, does the notation $k(E)$ means the number of connected components of G?
and about the definition of the connected component, i have got the following definition from wiki that a connected component 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. So according to this definition, is it make sense to say the complete graph $K_3$ has 2 connected component since $K_3$ itself have any two vertices connected; as well as another subgraph which by deleting one edge from $K_3$.
Best Answer
According to the final line in the definition, $k(E)$ is the number of connected components of the graph $(V,E)$. Since $G=(V,E)$, then $k(E)$ is the number of connected components of $G$.
The definition of a connected component in the first sentence of the Wikipedia article is somewhat confusing. The "As an equivalence relation" section has perhaps a better definition: say a vertex is reachable from another if there is a path from that vertex to the other. Reachability is an equivalence relation, and each equivalence class is called a connected component.
For example, since $K_n$ has a path from every vertex to every other, it has exactly one connected component. A visual way to think about connected components is that when you draw it on paper you can separate each connected component from each other, like in the illustration on the Wikipedia article.
Something to watch out for in the definition of $k$ is that $k(A)$ is the number of connected components of $(V,A)$, and not the number of connected components of the graph induced by the edges $A$ (this caught me when I first learned about the Tutte polynomial). For example, $k(\emptyset)=\lvert V\rvert$ since there are no paths from any vertex to any other, so each vertex is in its own connected component.