Question

Let G (V,E) be an undirected and unweighted graph with n = vertices and m edges. For a subset S CV, we can define a subgraph G' = (S, E') where E' CE and an edge (u, v) is included in E' if and only if u € S and v € S.

For any k < n, a graph is k-well-connected if every vertex has degree at least k. The degree of a vertex of a graph is the number of edges that are incident to the vertex.

Design an algorithm that runs in polynomial of n time that returns the set SCV of maximum size such that the subgraph G' = (S, E') is k-well-connected (k is given as input as a fixed value). You may assume that you can get the degree of a vertex in O(1) time and remove a vertex and its edges from the graph in O(degree of v) time.

Answer 1

The algorithm involves iteratively removing vertices that do not have a degree of at least k from the graph and their respective edges until all remaining vertices conform to the k-well-connected condition. This process is conducted in polynomial time, resulting in the largest set of vertices that induce a k-well-connected subgraph.

Step-by-step explanation:

The student's question is about designing an algorithm for identifying the largest subset of vertices in an undirected, unweighted graph such that the subgraph induced by those vertices is k-well-connected. We want this algorithm to run in polynomial time relative to the number of vertices, n. Here's a step-by-step polynomial-time algorithm to solve the problem:

1. Initialize set S to include all vertices of the graph G.
2. For each vertex v in S, check if the degree of v is at least k. If it is not, remove v from S and also remove all edges incident to v from graph G.
3. Repeat step 2 until all vertices in S have a degree of at least k.
4. Return the resulting set S as it represents the largest set of vertices such that the induced subgraph is k-well-connected.

This algorithm iteratively removes vertices not meeting the k-well-connected criteria, thus ensuring the final set S satisfies the required condition. Since the degree of each vertex can be checked in O(1) time and a vertex along with its edges can be removed in O(degree of v) time, these operations will not exceed polynomial time for all vertices.