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The 3-SAT problem can be reduced to the Hamiltonian cycle problem in polynomial time by representing each clause in the 3-SAT problem as a vertex in the Hamiltonian cycle graph, and connecting the vertices based on the relationships between the clauses. This reduction allows for solving the 3-SAT problem by finding a Hamiltonian cycle in the constructed graph.

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Q: How can the 3-SAT problem be reduced to the Hamiltonian cycle problem in polynomial time?
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How can the Hamiltonian cycle be reduced to a Hamiltonian path?

To reduce a Hamiltonian cycle to a Hamiltonian path, you can remove one edge from the cycle. This creates a path that visits every vertex exactly once, but does not form a closed loop like a cycle.


How can the Hamiltonian path be reduced to a Hamiltonian cycle?

To reduce a Hamiltonian path to a Hamiltonian cycle, you need to connect the endpoints of the path to create a closed loop. This ensures that every vertex is visited exactly once, forming a cycle.


How does the concept of a vertex cover relate to the existence of a Hamiltonian cycle in a graph?

In graph theory, a vertex cover is a set of vertices that covers all edges in a graph. The concept of a vertex cover is related to the existence of a Hamiltonian cycle in a graph because if a graph has a Hamiltonian cycle, then its vertex cover must include at least two vertices from each edge in the cycle. This is because a Hamiltonian cycle visits each vertex exactly once, so the vertices in the cycle must be covered by the vertex cover. Conversely, if a graph has a vertex cover that includes at least two vertices from each edge, it may indicate the potential existence of a Hamiltonian cycle in the graph.


What is the significance of a Hamiltonian cycle in a bipartite graph and how does it impact the overall structure and connectivity of the graph?

A Hamiltonian cycle in a bipartite graph is a cycle that visits every vertex exactly once and ends at the starting vertex. It is significant because it provides a way to traverse the entire graph efficiently. Having a Hamiltonian cycle in a bipartite graph ensures that the graph is well-connected and has a strong structure, as it indicates that there is a path that visits every vertex without repeating any. This enhances the overall connectivity and accessibility of the graph, making it easier to analyze and navigate.


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Yes. It is done by a technique called pulse width modulation. Similar to the principle used by incandescent lamp dimmers. Reduced duty cycle = reduced average current/power = reduced apparent brightness. Integrated circuits for this purpose are available from several sources.Artificial means of dimming LEDs also exist such as the patent-pending LEDdim dots and blocks.