An Eulerian cycle is a unidirectional cycle in which each edge appears precisely once. Create a graph client Euler that either discovers an Eulerian cycle or reports that none exist. Prove that a digraph G has a directed Eulerian cycle if and only if it is connected and each vertex's indegree equals its outdegree.

A graph is biconnected if every pair of vertices is connectedby two disjoint paths. An articulation point in a connected graph is a vertex that woulddisconnect the graph if it (and its adjacent edges) were removed. Prove that any graphwith no articulation points is biconnected. Hint : Given a pair of vertices s and t and apath connecting them, use the fact that none of the vertices on the path are articulationpoints to construct two disjoint paths connecting s and t.

Justify whether true or false:  For any undirected connected graph G, there exists at least two nodes such that removing thenodes won’t disconnect the graph.