Question

Write a polynomial-time algorithm that checks if an undirected graph has a Hamiltonian Circuit, assuming that the graph has no vertex with degree exceeding 2

Short answer

A polynomial-time algorithm can be implemented to check if an undirected graph has a Hamiltonian Circuit under the given conditions. This can be achieved by performing a depth-first or breadth-first search to determine if the graph is fully connected. If all vertices have been visited, the graph has a Hamiltonian Circuit.

Step-by-step solution

Initialize

This algorithm starts by choosing an arbitrary vertex as the initial vertex. Let's denote it as 'v'. You also need to initialize an empty set 'visited' to keep record of the visited vertices.

Traverse through graph

Now, traverse through the graph using a Depth-First Search (DFS) or Breadth-First Search (BFS) starting from 'v' and mark each traversed vertex as visited by adding them to the 'visited' set. It is important to note that DFS or BFS can reach a vertex in the graph if and only if it's connected to the starting vertex 'v'.

Check if graph is connected

Finally, check if all the vertices in the graph have been visited. If they have, the graph is connected, which means it has a Hamiltonian Circuit. If there are vertices that were not visited, these are connected to 'v' via a path, which means the graph is not connected and thus, does not have a Hamiltonian Circuit.

Key concepts

Polynomial-Time Algorithm

Understanding polynomial-time algorithms is crucial in recognizing the complexity of a problem-solving method. In the context of checking if an undirected graph contains a Hamiltonian Circuit, a polynomial-time algorithm is one where the time it takes to complete the problem is a polynomial function of the input size, commonly denoted as O(n^k) for some constant k. This designation is significant because polynomial-time algorithms are considered to be efficient and feasible for execution on realistic hardware, as opposed to exponential-time algorithms, which may become impractical even for moderately sized inputs.

In our exercise, the algorithm provided is polynomial because it involves a simple traversal of the graph, which can be done in linear or near-linear time depending on the graph's representation. The traversal step ensures that if all the vertices are visited, the graph is connected; this check is key in determining whether a Hamiltonian Circuit exists. Thus, the given algorithm performs in polynomial time, ensuring a quick and manageable solution to the problem.

Undirected Graph

An undirected graph is a fundamental concept in graph theory. It consists of a set of vertices (or nodes) and edges that connect pairs of these vertices. In contrast to directed graphs, where each edge has a direction, the edges in an undirected graph do not point any particular way. This means that if there is an edge between vertices A and B, you can travel from A to B and from B to A without any restriction.

Undirected graphs are pivotal in the study of Hamiltonian Circuits because these circuits seek a path that visits each vertex exactly once and ends at the originating vertex. The constraint in the given exercise—that no vertex has a degree (number of edges connected to it) exceeding 2—is a necessary condition for the existence of a Hamiltonian Circuit. Understanding the way edges and vertices are structured in an undirected graph assists in visualizing possible Hamiltonian Circuits and applying algorithms to find them.

DFS (Depth-First Search)

Depth-First Search, or DFS, is a classic graph traversal method that explores a graph by moving as far along a branch as possible before backtracking. The algorithm starts at a chosen vertex and explores as deep as it can along each branch before backtracking. This technique uses a stack to keep track of the path, which could be implemented either recursively or iteratively.

When applied to Hamiltonian Circuit problems, DFS aids in efficiently exploring all possible paths through a graph. A key characteristic of DFS is its ability to dive deep into the graph, potentially reaching all vertices in a connected component. It's well-suited for our exercise since we need to check if it's possible to visit all vertices once. If our DFS traversal marks all vertices as visited, then the graph is connected and could support a Hamiltonian Circuit, given the degree constraint is also met.

BFS (Breadth-First Search)

Breadth-First Search, or BFS, is another fundamental graph traversal algorithm, which operates on a different principle than DFS. BFS explores the graph level by level, starting from a chosen vertex and visiting all its direct neighbors before moving on to their unvisited neighbors. This process continues until all vertices in the graph have been visited. A queue typically facilitates the BFS algorithm to manage the vertices to be explored.

In the scope of our exercise, BFS can be employed as an alternative to DFS for checking the connectedness of the graph. Since BFS explores neighbors before proceeding to the next level, it can quickly determine if every vertex is reachable from a starting point. The level-by-level approach of BFS provides a complementary strategy to DFS. However, it's important to note that for the given problem, whether you choose BFS or DFS, the outcome will be the same in terms of detecting if a Hamiltonian Circuit exists.