Can a disconnected graph have an Eulerian cycle?

An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected.

Can every connected graph be Eulerian?

Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. If there are no vertices of odd degree, all Eulerian trails are circuits.

Can a graph have a disconnected vertex?

An undirected graph that is not connected is called disconnected. An undirected graph G is therefore disconnected if there exist two vertices in G such that no path in G has these vertices as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.

Is a graph with one vertex Eulerian?

If a graph consists of a single vertex v, then the path consisting of v is vacuously Eulerian. It is also a Hamiltonian path, since it contains all of the vertices of the graph.

Which is a cut vertex in a graph?

Articulation Points (or Cut Vertices) in a Graph. A vertex in an undirected connected graph is an articulation point (or cut vertex) iff removing it (and edges through it) disconnects the graph. Articulation points represent vulnerabilities in a connected network – single points whose failure would split the network into 2 or more components.

How to prove that an Eulerian graph cannot have a minimal edge cut?

3. Prove that an Eulerian graph cannot have a minimal edge cut with an odd number of edges. (4) Consider any minimal edge cut X. Then, G – X has exactly two com- ponents (or X will not be minimal). Since G is Eulerian, G can be decomposed into cycle. If a cycle lies entirely in one component, no edge of the cycle belongs to X.

When does a graph become an Euler circuit?

An Euler circuit always starts and ends at the same vertex. A connected graph G is an Euler graph if and only if all vertices of G are of even degree, and a connected graph G is Eulerian if and only if its edge set can be decomposed into cycles.

What makes a vertex an articulation point in a graph?

Last Updated : 25 Mar, 2021 A vertex in an undirected connected graph is an articulation point (or cut vertex) iff removing it (and edges through it) disconnects the graph. Articulation points represent vulnerabilities in a connected network – single points whose failure would split the network into 2 or more components.