Every pair of vertices in a graph that is linked by two different paths is said to be biconnected. An articulation point in a connected graph is a vertex that, if it and its surrounding edges were eliminated, would cause the graph to become disconnected. demonstrate the biconnection of any graph lacking articulation points. Tip: To create two disjoint paths connecting s and t given a set of vertices s and t and a path connecting them, take advantage of the fact that none of the vertices on the path are articulation points.

Give an example of a graph that has all 3 of the following properties. (Note that you need to give a single graph as the answer.) (i)  It is connected (ii)  It has one articulation point. (iii)  The graph needs at least 4 colors for a valid vertex coloring (iv)  The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.

2. Answer the following questions about the graph given below. B E a) Does an Euler circuit exist? Why or why not? b) Find a good eulerization for the graph, using as few duplicated edges as possible. How many edges must be added? c) Use the eulerization you developed in part b) to determine a best circuit for this graph, beginning at vertex A. List this path by listing successive vertices. 1 3. Consider the complete graph given below. 24 14 46 B, 32 51 28 a) Find the length of the sorted-edges tour for this graph. Use wiggly lines to indicate this tour. b) Find the length of the nearest-neighbor tour starting at vertex A. Use wiggly lines to indicate this tour.

Write a function that returns true (1) if there is an edge between two vertices u and v in an adjacency-listed directed graph.