## Can directed graphs have cycles?

A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree.

**How do you find all cycles in a directed graph?**

To detect cycle, check for a cycle in individual trees by checking back edges. To detect a back edge, keep track of vertices currently in the recursion stack of function for DFS traversal. If a vertex is reached that is already in the recursion stack, then there is a cycle in the tree.

**How many cycles can a directed graph have?**

If you graph sin(x) from 0 to 360 degrees, you will get one cycle, but if you think about the graph, f(x) = sin(x), from -∞ to +∞, there will be an infinite number of cycles.

### What is the chromatic number of CN?

Theorem 9: The chromatic number of Cn is 2 if n is even, and 3 if n is odd. Proof: First note that the chromatic number must be at least 2 for any graph which has an edge in it, including all cycles.

**How do you remove cycles from a graph?**

One way to do this is simply drop edges from the task graph to break the cycle. A feedback arc set or feedback edge set is a set of edges which when removed from the graph will leave a DAG. Put another way, it is a set containing at least one edge of every cycle in the graph.

**Which algorithm can be used for cycle detection algorithm in directed graph?**

In my opinion, the most understandable algorithm for detecting cycle in a directed graph is the graph-coloring-algorithm. Basically, the graph coloring algorithm walks the graph in a DFS manner (Depth First Search, which means that it explores a path completely before exploring another path).

#### How many cycles are there in a graph?

**How many cycles are in a complete graph?**

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

**What is vertex coloring of a graph?**

A vertex coloring is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. The most common type of vertex coloring seeks to minimize the number of colors for a given graph.

## What makes a Euler circuit?

An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler path starts and ends at different vertices. ▶ An Euler circuit starts and ends at the same vertex.

**Does Kruskal algorithm work for directed graphs?**

No, Prim’s and Kruskal’s algorithm works only for undirected graphs. For directed graphs, the equivalent notion of a spanning tree is spanning arborescence. A minimum weight spanning arborescence can be found using Edmonds’ algorithm.

**Can BFS detect cycle?**

BFS wont work for a directed graph in finding cycles. Consider A->B and A->C->B as paths from A to B in a graph. BFS will say that after going along one of the path that B is visited. When continuing to travel the next path it will say that marked node B has been again found,hence, a cycle is there.