How many edges can be removed from a connected graph?

How many edges can be removed from a connected graph?

m−n+1 edges need to be removed.

What is the minimum number of edges needed to make the graph connected?

n-1
The minimum number of edges for undirected connected graph is (n-1) edges. To see this, since the graph is connected then there must be a unique path from every vertex to every other vertex and removing any edge will make the graph disconnected.

Which edge removal makes the graph disconnected?

cut edge
An edge ‘e’ ∈ G is called a cut edge if ‘G-e’ results in a disconnected graph. If removing an edge in a graph results in to two or more graphs, then that edge is called a Cut Edge. In the following graph, the cut edge is [(c, e)]. By removing the edge (c, e) from the graph, it becomes a disconnected graph.

How many edges would we have to delete from this undirected graph to make it a tree?

Removing any one of the edges will make the graph acyclic. Therefore, at least one edge needs to be removed. Explanation: Graph is already acyclic. Therefore, no edge removal is required.

What is the maximum number of edges in an undirected graph with V vertices?

The maximum number of edges in an undirected graph is n(n-1)/2 and obviously in a directed graph there are twice as many. If the graph is not a multi graph then it is clearly n * (n – 1), as each node can at most have edges to every other node. If this is a multigraph, then there is no max limit.

How do you tell if a graph is connected or disconnected?

A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected.

Can a simple graph be disconnected?

A simple graph, also called a strict graph (Tutte 1998, p. 2), is an unweighted, undirected graph containing no graph loops or multiple edges (Gibbons 1985, p. A simple graph may be either connected or disconnected. Unless stated otherwise, the unqualified term “graph” usually refers to a simple graph.

How to reduce the number of crossing edges in a graph?

Here is an idea to get around the issues concerning the shape and size of nodes. Given the graph (infinitely small nodes), expand each node while “pushing” or bending edges out of the way (ex. using splines while enforcing a limit on proximity).

What’s the maximum number of edges you can remove?

Removing any additional edge will not make it so. So the maximum number of edges we can remove is 2. Input: n = 4, edges = [ [3,1,2], [3,2,3], [1,1,4], [2,1,4]] Output: 0 Explanation: Notice that removing any edge will not make the graph fully traversable by Alice and Bob.

When does a graph have more than one edge?

This graph has more edges, contradicting the maximality of the graph. Hence the maximum is achieved when only one of the components has more than one vertex. How many vertices does this graph have? the big component has n − k + 1 vertices and is the only one with edges.

Which is the maximum number of edges in ith component G?

Thus the required inequality is proved. Now the maximum number of edges in ith component of G (which is simple connected graph) is 1 2ni(ni − 1). Therefore, the maximum number of edges in G is I’ve answered the OP’s specific question as to how the book’s proof makes sense.