Consider the graph G = , with, V = {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q}.

W = {(A,B), (A,C), (A,D), (B,C), (C,D),(D,E), (E,F), (E,H), (E,I), (F,G), (F,H), (F,I), (G,H), (G,I) (I, J), (J,K), (J,L), (J,M), (J,N), (J,Q), (K,L), (K,M), (K,N), (K,Q), (L,M), (L,N), (L,Q), (M,N), (M,Q), (N,Q)}

1. Considering G as an undirected graph, do the following problems:

a) Draw the graph G, (give a visual representation).

b) Give the Adjacency List representation of G.

c) Give a linked-list representation of the Adjacency List of G.

d) Give the Adjacency Matrix representation of G.

e) Does G have any Isolated node? Explain. (x2) How can this be determined using the Adjacency List?

f) Is G connected? (x2) Explain.

g) List the nearest, 1-hop neighbors of node F.

h) What is a Clique in a graph? (x2) Identify as many cliques as you can in G.

i) An Articulation Point of G is a node or vertex whose removal disconnects G. Identify the articulation points, if any of G. (x2) Which graph representation is the easiest to use to make this determination?

j) A Bridge of G is an edge whose removal disconnects G. Identify the bridges, if any in G.

k) Trace a Path (or route) from node A to node Q.

l) Modify G into a new graph H, by adding the edges {(I,K), (I,N), (C,E), (D,G), (K,O), (K,P), (M,O), (P,Q)}.

m) Show a Breadth-First Search (BFS) traversal (tour) of H, starting from node A.

n) Show a Depth-First Search (DFS) traversal (tour) of H, starting from node K.

o) A Spanning Tree of a graph is a sub-graph which includes all the nodes of the graph and the sub-graph also forms a tree. (x1) Is the sub-graph formed by the BFS in (1.k) a spanning tree? Explain. (x2) Is the sub-graph formed by the DFS in (1.l) a spanning tree? Explain.

p) Give a binary search tree representation of the edge list of node K in the graph H.

4.

a. Is the graph G of question (1), planar or non-planar? Explain.

b. The incidence matrix Z of a simple digraph d = is the |v| x |e| matrix Z = (zjk), such that Zjk = -1 if edge k leaves node j; Zjk = 1 if edge k enters node j; Zjk = 0, otherwise. (x1) Give the incidence matrix Z of the digraph R of (3.d). (x2) Describe what the entries of the matrix product ZZ represent, where Z is the transpose of Z.

Please do question 1) g through p and question 4.