Question

How many edges are in the transitive closure of a graph that consists of a simple directed path of n vertices?

Short answer

The transitive closure has \[ \frac{(n-1)n}{2} \] edges.

Step-by-step solution

Understand the Transitive Closure

The transitive closure of a graph is a new graph that contains an edge from vertex A to vertex B if there is a path (of any length) from A to B in the original graph.

Graph Structure

A simple directed path of n vertices means the vertices are arranged in a linear order where each vertex points to the next one. For example, for 4 vertices it would look like: 1 -> 2 -> 3 -> 4.

Identify All Possible Paths

In the transitive closure, every vertex will be connected to every other vertex that comes after it in the sequence. So, for vertex 1, there will be edges to vertices 2, 3, 4, ..., n. For vertex 2, there will be edges to vertices 3, 4, ..., n, and so on.

Count the Edges

The number of edges from vertex i to any vertex after it is n - i. Summing this for all vertices, the total number of edges will be \[ (n-1) + (n-2) + ... + 1 \]. This can be computed using the formula for the sum of the first (n-1) natural numbers: \[ \sum_{i=1}^{n-1} i = \frac{(n-1)n}{2} \].

Final Calculation

Using the formula, the total number of edges in the transitive closure is \[ \frac{(n-1)n}{2} \].

Key concepts

Directed Graphs

In graph theory, a directed graph (or digraph) is a set of vertices connected by edges, where the edges have a direction. This means every edge goes from one vertex to another specified vertex and does not necessarily go both ways. These graphs are easy to visualize using arrows that indicate the direction of each edge. For example, in a directed graph with 4 vertices, you might have connections like 1 -> 2, 2 -> 3, and 3 -> 4. This kind of arrangement where each vertex points to the next one creates a simple directed path.

Path Counting

Path counting in a graph involves determining the number of distinct paths between vertices. In the context of transitive closure, we look at all possible connections. For a simple directed path of n vertices, every vertex is connected to every vertex that comes after it. For vertex 1, there will be direct paths to vertices 2, 3, 4, ..., n. For vertex 2, there will be paths to vertices 3, 4, ..., n, and so forth. This counting is crucial because it helps in understanding the transitive closure, where indirect paths are converted into direct edges.

Edge Calculation

Calculating the number of edges in the transitive closure involves summing up all the possible directed connections in the graph. In the transitive closure of a simple directed path with n vertices, each vertex is connected to all subsequent vertices. For vertex i, the total number of edges connecting it to later vertices is given by n - i. Summing these for all vertices, the total number of edges can be determined using the formula for the sum of the first (n-1) natural numbers: For a path with n vertices, this sum is: Summing up these edges, we get: That means, the total number of edges in the transitive closure of a simple directed path with n vertices is So, in a linear path with n vertices, each previous vertex will be connected to the following vertices in sequence, leading to this calculated closure.