Chapter 9: Problem 3

Show that a graph problem using the number of vertices as the measure of the size of an instance is polynomially equivalent to one using the number of edges as the measure of the size of an instance.

Short Answer

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A graph problem using the number of vertices as the measure of the size of an instance is polynomially equivalent to one using the number of edges as the measure of the size of an instance. The reason being the polynomial relationship between the number of edges and number of vertices as per graph theory fundamentals.

Step by step solution

Understand the basics of a Graph

In a graph, vertices (or nodes) are the fundamental units which are connected by edges (or arcs). The total number of edges can range anywhere between 1 and \((n*(n-1))/2\), where n is the number of vertices, in a complete graph. The minimum number of edges is 1 indicating a disconnected graph while the latter indicates a fully connected graph.

Define the Polynomial Equivalence

In computational theory, two measures are polynomially equivalent if there exists a polynomial function that translates the magnitude of one measure into the magnitude of the other. In simpler terms, if the measure of one instance increases, the measure of the other instance will also increase at a rate that can be expressed as a polynomial function. It symbolizes that both measures grow at an approximately similar pace.

Reason the Polynomial Equivalence between Variations in Vertices and Edges

The change in the number of edges has a squared relationship with the change in the number of vertices. This is because when we add a vertex, we have the potential to add an edge for each existing vertex. Therefore, we have a relationship of \( n^{2} \), where n is the number of vertices. This indicates polynomial equivalence as the square of n is a polynomial function, and a change in the number of vertices causes a change in the number of edges that follows this polynomial function.

Draw the Conclusion

Thus, in a graph problem, a graph with vertices as the measure of the size of an instance is polynomially equivalent to one with the number of edges as the measure of the size of an instance, as both measures are interconnected and grow in a polynomial manner relative to each other. This relationship holds true for all types of graph, whether connected, disconnected, sparse or dense.

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Show that a graph problem using the number of vertices as the measure of the size of an instance is polynomially equivalent to one using the number of edges as the measure of the size of an instance.

Short Answer

Step by step solution

Understand the basics of a Graph

Define the Polynomial Equivalence

Reason the Polynomial Equivalence between Variations in Vertices and Edges

Draw the Conclusion

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