Time Complexity and graphs

Question

I'm learning graphs these days and need to clear few doubts-

1. Can I determine weather 5 points in two dimensions whose X and Y coordinates are given lie on the same straight line in O(1).
2. What is the efficient way to find out the degree of the specified vertex in an undirected graph (V,E)? I'm thinking of O(|E|)?
3. What is the efficient way to find degrees of all vertices in an undirected graph (V,E)? I'm think of O(|V||E|)
4. if |E| ≥ |V| can I say grapgh will be have guaranteed Euler circuit?
5. Is it possible to find slope of a straight line in O(1)time given the X and Y coordinates of n points on the line?

Answer 1

1. Yes, anything that has constant input could be done in $O(1)$. Howether if the coordinates have size more than $O(1)$ than you will not be able to even read them in time.
2. That depends on your representation of the graph and what operations do you count. If you assume that you read the whole graph in some way and then find the needed degree, the complexity will be $O(|V| + |E|)$ if your representation is an edge list, $O(|V|^2)$ if it is an adjacency matrix. If you assume that you already have some representation of the graph in memory you can find the degree in $O(|V|)$ if you have the adjacency matrix.
3. You can do it in $O(|V| + |E|)$, start with an empty graph (all the degrees are zero) and add the edges one by one. When you add an edge to the graph only two vertices change their degrees.
4. No, consider the graph $K_4$ (complete graph on $4$ vertices). All the degrees in it are odd so it does not have even en Euler path.
5. If you want to read the coordinates it takes $O(n)$ operations, so no.

Answer 2

Just adding to what @Artur said: You only need 2 points to find the slope of a straight line, even if you're given more, so that's O(1) for me...