Question

Create a directed graph G with a DFF that contains a single vertex v as one of its trees, even though both the indegree and outdegree of v are greater than 1 in G. State the DFS vertex order that causes the vertex to be isolated in the DFF.

Answer 1

A directed graph G with a DFF that contains a single vertex v as one of its trees, even though both the indegree and outdegree of v are greater than 1 in G can be created by introducing a self-loop on the vertex v. By doing this, v becomes isolated in the DFF because the self-loop allows v to be its own parent, thereby creating a tree with only v.

Step-by-step explanation:

A directed graph G with a DFF that contains a single vertex v as one of its trees, even though both the indegree and outdegree of v are greater than 1 in G can be created by introducing a self-loop on the vertex v. This means that v has an edge that starts and ends at itself. By doing this, v becomes isolated in the DFF because the self-loop allows v to be its own parent, thereby creating a tree with only v.

For example, consider the directed graph G with vertex v and self-loop edge (v,v). In this case, the indegree and outdegree of v are both 2, but v is isolated in the DFF because it forms a tree by itself.

The DFS (Depth First Search) vertex order that causes the vertex v to be isolated in the DFF depends on the implementation of the algorithm. However, one possible DFS vertex order that isolates v in the DFF is: v.