Final answer:
The star-search problem is a computational complexity problem in mathematics that involves determining the existence of a particular configuration of stars in a given graph. To show that it is NP-complete, we can demonstrate that it is in the class NP and that it is also NP-hard by reducing a known NP-complete problem, such as the vertex cover problem, to the star-search problem.
Step-by-step explanation:
The star-search problem is a problem in computational complexity theory that involves determining the existence of a particular configuration of stars in a given graph. To show that the star-search problem is NP-complete, we need to demonstrate that it is both in the class NP and that it is NP-hard. One way to show this is by reducing a known NP-complete problem, such as the vertex cover problem, to the star-search problem.
First, we need to show that the star-search problem is in NP. This means that given a potential solution to the problem, we can verify its correctness in polynomial time. In the case of the star-search problem, a potential solution would be a set of stars in the graph that satisfy the given criteria. We can easily check if this set of stars form a valid configuration in polynomial time.
Next, we need to prove that the star-search problem is NP-hard by reducing the vertex cover problem to it. The vertex cover problem asks whether a given graph has a set of vertices such that every edge in the graph is incident to at least one vertex in the set. We can construct an instance of the star-search problem by representing the edges of the graph as stars and the vertices of the graph as nodes. The height of each star is set to 1, and the distance between two stars is set to 2 if the corresponding vertices are adjacent in the graph. By transforming the vertex cover problem to the star-search problem in this way, we can show that the star-search problem is NP-hard.