Let G be a simple undirected graph. Let TD be a depth first search tree of G. Let TB be a breadth first search tree of G. Consider the following statements.
(I) No edge of G is a cross edge with respect to TD. (A cross edge in G is between two nodes neither of which is an ancestor of the other in TD).
(II) For every edge (u, v) of G, if u is at depth i and v is at depth j in TB, then ∣i − j∣ = 1.
Which of the statements above must necessarily be true?
(A) I only
(B) II only
(C) Both I and II
(D) Neither I nor II
Explanation: There are four types of edges can yield in DFS. These are tree, forward, back, and cross edges. In undirected connected graph, forward and back egdes are the same thing. A cross edge in a graph is an edge that goes from a vertex v to another vertex u such that u is neither an ancestor nor descendant of v. Therefore, cross edge is not possible in undirected graph.
So, statement (I) is correct.
For statement (II) take counterexample of complete graph of three vertices, i.e., K3 with XYZ, where X is source and Y and Z are in same level. Also,there is an edge between vertices Y and Z, i.e., |i-j| = 0 ≠ 1 in BFS. So, statement became false.
Option (A) is correct.
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