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1

### GATE CSE 2007

What is the largest integer m such that every simple connected graph with n vertices and n edges contains at least m different spanning trees?
A
1
B
2
C
3
D
n
2

### GATE CSE 2006

Let T be a depth-first search tree in an undirected graph G. Vertices u and v are leaves of this tree T. The degrees of both u and v in G are at least 2. Which one of the following statements is true?
A
There must exist a vertex w adjacent to both u and v in G
B
There must exist a vertex w whose removal disconnects u and v in G
C
There must exist a cycle in G containing u and v
D
There must exist a cycle in G containing u and all its neighbors in G.
3

### GATE CSE 2006

Let T be a depth first search tree in an undirected graph G. Vertices u and v are leaves of this tree T. The degrees of both u and v in G are at least 2. which one of the following statements is true?
A
There must exist a vertex w adjacent to both u and v in G
B
There must exist a vertex w whose removal disconnects u and v in G
C
There must exist a cycle in G containing u and v
D
There must exist a cycle in G containing u and all its neighbours in G.
4

### GATE CSE 2006

Consider the depth-first-search of an undirected graph with 3 vertices P, Q, and R. Let discovery time d(u) represent the time instant when the vertex u is first visited, and finish time f(u) represent the time instant when the vertex u is last visited. Given that
d(P) = 5 units      f(P) = 12 units
d(Q) = 6 units      f(Q) = 10 units
d(R) = 14 unit      f(R) = 18 units
Which one of the following statements is TRUE about the graph
A
There is only one connected component
B
There are two connected components, and P and R are connected
C
There are two connected components, and Q and R are connected
D
There are two connected components, and P and Q are connected
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