 # Question: Can Undirected Graphs Have Cycles?

## Can we use BFS to detect cycle in an undirected graph?

Like directed graphs, we can use DFS to detect cycle in an undirected graph in O(V+E) time.

We do a BFS traversal of the given graph.

For every visited vertex ‘v’, if there is an adjacent ‘u’ such that u is already visited and u is not parent of v, then there is a cycle in graph..

## How do you prove a graph has a cycle?

Proof: Let G be a graph with n vertices. If G is connected then by theorem 3 it is not a tree, so it contains a cycle. If G is not connected, one of its connected components has at least as many edges as vertices so this component is not a tree and must contain a cycle, hence G contains a cycle.

## What is a cycle in an undirected graph?

There is a cycle in a graph only if there is a back edge present in the graph. … To find the back edge to any of its ancestor keep a visited array and if there is a back edge to any visited node then there is a loop and return true. Algorithm: Create the graph using the given number of edges and vertices.

## Is a self loop a cycle?

A self-loop or loop is an edge between a vertex and itself. An undirected graph without loops or multiple edges is known as a simple graph. … A cycle is a closed path, i.e. a path combined with the edge (vk,v1). A graph is connected if there exists a path between each pair of vertices.

## How many cycles does a graph have?

Actually a complete graph has exactly (n+1)! cycles which is O(nn).

## Does a loop count as an edge?

In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops.

## Is a loop 2 Edges?

In other words, a loop is an edge that starts and finishes in the same node. That node is connected to itself, and therefore is its own neighbor. You can also see that nodes 1 and 3 are connected by two edges. Those edges are “parallel edges”, or “multiple edges”.

## What is a positive length cycle?

A cycle is a positive length closed walk whose vertices are distinct except for the beginning and end vertices. Note that a single vertex counts as a length zero path that begins and ends at itself. … Length one cycles are possible when a node has an arrow leading back to itself.

## Can topological sort detect cycles?

An interesting and simple way to find the existence of cycles in a directed graph is using Kahns algorithm for topological sort.

## How do you determine if an undirected graph has a cycle?

To detect if there is any cycle in the undirected graph or not, we will use the DFS traversal for the given graph. For every visited vertex v, when we have found any adjacent vertex u, such that u is already visited, and u is not the parent of vertex v. Then one cycle is detected.

## Does a loop count as a degree?

…with each vertex is its degree, which is defined as the number of edges that enter or exit from it. Thus, a loop contributes 2 to the degree of its vertex.

## What is a cycle graph theory?

In graph theory, a cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices. … A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree.

## How many edges does an undirected graph have?

An undirected graph has no directed edges. Consider the following examples. In the above Undirected Graph, deg(a) = 2, as there are 2 edges meeting at vertex ‘a’.

## Can undirected graphs have self loops?

In particular, unless otherwise specified, a graph will refer to a simple undirected graph: an undirected graph where each edge connects two distinct vertices (thus no self-loops) and there is at most one edge between each pair of vertices (no parallel edges).

## What is undirected graph?

An undirected graph is graph, i.e., a set of objects (called vertices or nodes) that are connected together, where all the edges are bidirectional. An undirected graph is sometimes called an undirected network. In contrast, a graph where the edges point in a direction is called a directed graph.

## What is difference between directed and undirected graph?

Undirected graphs have edges that do not have a direction. The edges indicate a two-way relationship, in that each edge can be traversed in both directions. … Directed graphs have edges with direction. The edges indicate a one-way relationship, in that each edge can only be traversed in a single direction.

## Does a graph have to be connected?

A graph is connected if and only if it has exactly one connected component. The strong components are the maximal strongly connected subgraphs of a directed graph. A vertex cut or separating set of a connected graph G is a set of vertices whose removal renders G disconnected.

## Can a simple graph have cycles?

A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex. In other words a simple graph is a graph without loops and multiple edges. … A circuit that doesn’t repeat vertices is called a cycle.