Advanced Graph Algorithms and Structures Quiz Quiz

Explore challenging concepts in advanced graph algorithms and data structures, covering topics like spanning trees, network flows, planarity, and graph isomorphism. This quiz is designed for learners seeking to deepen their understanding of graph theory and its algorithmic techniques.

1. Minimum Spanning Tree Algorithms

   Which algorithm efficiently finds a Minimum Spanning Tree in a connected, weighted, undirected graph with distinct edge weights, and always produces the same unique tree for each input?

   1. Kruskal's algorithm
   2. Depth-First Search
   3. Floyd-Warshall algorithm
   4. Bellman-Ford algorithm

   Explanation: Kruskal's algorithm efficiently finds a Minimum Spanning Tree (MST) by building the tree edge by edge in increasing order of weight and avoids cycles. As the graph has distinct weights, the MST is unique, and Kruskal's algorithm always produces it. Depth-First Search is used for traversing trees or graphs and does not find MSTs. Bellman-Ford computes shortest paths from a single source, while Floyd-Warshall finds all-pairs shortest paths; neither directly serves MST construction.

2. Max-Flow Problem

   In the context of the maximum flow problem, which theorem guarantees that the value of the maximum flow from source to sink equals the capacity of the minimum cut in a flow network?

   1. Planarity Theorem
   2. Max-Flow Min-Cut Theorem
   3. Euler's Theorem
   4. Dijkstra's Principle

   Explanation: The Max-Flow Min-Cut Theorem establishes that in a flow network, the maximum amount of flow that can be sent from source to sink equals the smallest total capacity across all possible cuts separating them. Euler's Theorem deals with Eulerian paths and circuits. Dijkstra's Principle relates to shortest paths, and the Planarity Theorem describes when graphs can be drawn without edge crossings, none of which address flow or cuts.

3. Graph Planarity

   A graph is considered planar if and only if it does not contain a subgraph homeomorphic to which pair of graphs, according to Kuratowski's Theorem?

   1. K5 and K3,3
   2. K2,2 and K4,4
   3. C4 and K4
   4. Petersen and K3

   Explanation: Kuratowski's Theorem states that a graph is planar exactly when it contains no subgraph that is a subdivision (homeomorphic image) of K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph on two sets of three vertices). C4 and K4 are unrelated to planarity obstructions. Petersen and K3 are not the forbidden graphs for planarity. K2,2 and K4,4 do not characterize planarity in Kuratowski's theorem.

4. Graph Isomorphism Problem

   Given two graphs G and H, which of the following best describes the Graph Isomorphism problem?

   1. Determining if G and H are structurally identical up to relabeling of vertices
   2. Calculating the chromatic number of H
   3. Checking if G is a subgraph of H
   4. Finding the minimum spanning tree for both G and H

   Explanation: The Graph Isomorphism problem is to decide whether there is a bijection between the vertex sets of G and H such that adjacency is preserved, meaning the graphs are identical except for vertex labels. Finding a minimum spanning tree pertains to edge weights, not isomorphism. Checking if G is a subgraph of H is a different question, and computing the chromatic number relates to coloring, not structural identity.

5. Tree Decomposition and Width

   Which term describes the smallest width among all possible tree decompositions of a graph, providing a measure of how close the graph is to being a tree?

   1. Treewidth
   2. Node span
   3. Branchweight
   4. Loop number

   Explanation: Treewidth is the minimum width across all tree decompositions of a graph and quantifies how tree-like a graph is. Branchweight is not a standard term in this context and may be confused with edge weights. Node span and loop number are incorrect and not used for this purpose. Only treewidth accurately captures the required measure for tree decomposition.