Advanced Graph Theory: Bridges and Articulation Points Quiz Quiz

Challenge your understanding of advanced graph concepts with this focused quiz on bridges and articulation points. Evaluate your ability to identify critical edges and vertices in undirected graphs, ideal for students and enthusiasts of graph algorithms.

1. Definition of Bridges

   Which of the following best describes a bridge in an undirected graph?

   1. A vertex that connects two cycles together.
   2. A loop that connects a vertex to itself.
   3. An edge whose removal increases the number of connected components.
   4. A group of vertices with even degrees.

   Explanation: A bridge is defined as an edge that, when removed, causes the graph to become more disconnected by increasing its number of connected components. A vertex that connects two cycles is not a standard definition related to bridges. A loop does not affect connectivity between different vertices. The degree of a vertex is unrelated to the concept of bridges.

2. Identification of Articulation Points

   In a connected undirected graph, which statement about an articulation point is true?

   1. It is a vertex with the highest degree.
   2. It is any vertex present in a cycle.
   3. It is a vertex whose removal disconnects the graph.
   4. It is a vertex connected to all other vertices.

   Explanation: An articulation point is a vertex that, if removed along with its incident edges, increases the number of connected components in the graph. The highest degree vertex is not necessarily an articulation point. Not all vertices connected to all others (universal vertices) are articulation points. Vertices in a cycle may or may not be articulation points, depending on the structure.

3. Bridge Example Scenario

   Consider a path graph with vertices A-B-C. Which edge is a bridge?

   1. Only edge AC.
   2. Both edges AB and BC.
   3. Edge BA only.
   4. No edge in the graph.

   Explanation: In a simple path A-B-C, both AB and BC are bridges because removing either edge will disconnect the graph. Edge AC does not exist in the path. There is no edge BA unless the graph is undirected, but as the graph is undirected, AB and BA refer to the same edge. Saying 'no edge in the graph' is incorrect because there are bridges present.

4. Root in DFS and Articulation Points

   What special case applies to the root of a DFS tree when identifying articulation points?

   1. The root is an articulation point only if it is a leaf.
   2. The root is an articulation point only if it has more than one child.
   3. The root is always an articulation point.
   4. The root cannot be an articulation point.

   Explanation: During DFS, the root is an articulation point only if it has two or more children because its removal would disconnect those subtrees. The root is not automatically an articulation point, contrary to always being so. If the root is a leaf, removing it does not disconnect the graph. The root can be an articulation point, contradicting the option that says it cannot.

5. Finding Bridges Efficiently

   Which algorithm is commonly used to find bridges in an undirected graph in linear time?

   1. Dijkstra’s shortest path algorithm.
   2. Depth-first search (DFS) with discovery and low-link values.
   3. Prim’s minimum spanning tree algorithm.
   4. Floyd-Warshall’s all-pair shortest paths.

   Explanation: DFS with discovery and low-link values efficiently finds all bridges in O(V+E) time, tracking the earliest visited vertex reachable from each subtree. Dijkstra’s and Floyd-Warshall algorithms are for shortest paths, not bridge detection. Prim’s algorithm constructs a minimum spanning tree and is unrelated to bridge identification.

6. Impact of Bridge Removal

   What effect does removing a bridge from a connected undirected graph have?

   1. It creates a new cycle in the graph.
   2. It reduces the degree of every vertex by one.
   3. It increases the number of connected components.
   4. It always reduces the number of edges to half.

   Explanation: Removing a bridge disconnects two parts of the graph, increasing the number of connected components. Removing a bridge does not create cycles; in fact, it breaks paths. The degree reduction only applies to the two vertices incident to the bridge, not all vertices. Bridge removal does not guarantee halving the number of edges.

7. Single-Point Failure

   If a network’s topology has one articulation point, what can be said about its fault tolerance?

   1. There are no bridges in the network.
   2. Every vertex is an articulation point.
   3. The network cannot be disconnected by removing any one vertex.
   4. The articulation point is a single point of failure.

   Explanation: An articulation point is a vulnerability in the network; its removal disconnects the network, making it a single point of failure. Claiming no single vertex disconnects the network is incorrect as defined. The presence of an articulation point does not necessarily mean there are no bridges, and not every vertex in such a network will be an articulation point.

8. Bridge in a Cycle

   In a simple cycle graph with four vertices, are there any bridges present?

   1. Yes, all edges are bridges in a cycle.
   2. Exactly two edges are bridges.
   3. No, because removal of any edge does not disconnect the graph.
   4. Only the edge connecting the first and last vertices is a bridge.

   Explanation: A cycle remains connected if any single edge is removed; each vertex can still reach the others through alternate paths. Therefore, none of these edges are bridges. If every edge were a bridge, the removal of any single edge would disconnect the graph, which is false for cycles. The concept of 'first and last vertices' in a cycle does not apply, and only two bridges are not present in a four-vertex cycle.

9. Identifying Articulation Points in Trees

   Which statement about articulation points in a tree with more than two vertices is correct?

   1. Every non-leaf vertex is an articulation point.
   2. All vertices are articulation points.
   3. Only leaf nodes are articulation points.
   4. There are no articulation points.

   Explanation: In a tree, the removal of any non-leaf vertex disconnects its connected subtrees, making every non-leaf an articulation point. There are articulation points in trees with more than two vertices, refuting the claim of none. Leaf nodes are not articulation points since removing them does not disconnect the tree. Not all vertices, especially leaf nodes, are articulation points.

10. Minimum Number of Bridges

    What is the minimum number of bridges in any connected tree with N vertices?

    1. N-1
    2. N
    3. 1
    4. 0

    Explanation: A tree with N vertices has exactly N-1 edges, and since the removal of any edge disconnects the tree, every edge is a bridge. The minimum is not 1 except in a two-vertex tree, but generally, it is N-1. Saying there are zero bridges is incorrect since every edge is critical. N is not the right answer as the tree has N-1 edges, not N.