In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

1. v₁e₂v₂e₃v₃e₄v₄e₅v₂e₂v₁e₁v₀
2. v₂v₃v₄v₅v₂
3. v₄ v₂ v₃ v₄ v₅ v₂ v₄
4. v₂ v₁ v₅ v₂ v₃ v₄ v₂
5. v₀v₅v₂v₃v₄v₂v₁
6. v₅v₄v₂v₁

(a)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

Given information:

Calculation:

We know that:

A walk from v to w is a finite sequence of alternatively adjacent vertices and edges of a graph.

A trail from v to w is a walk that does not contain a repeated edge.

A path from v to w is a trial that does not contain a repeated vertex.

A closed walk is a walk that starts and ends at the same vertex.

A circuit is a closed walk with at least one edge and no repeated edges.

A simple circuit is a circuit with no repeated vertices except for the first and last vertex.

We have: v1e2v2e3v3e4v4e5v2e2v1e1v0.

The walk v1e2v2e3v3e4v4e5v2e2v1e1v0 is not a trial, because we note that the walk contains the edge e₂ twice.

The walk v1e2v2e3v3e4v4e5v2e2v1e1v0 is not a path, because the walk is not a trial

(b)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

(c)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

(d)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

(e)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.

(f)

In the graph below, determine whether the following walks are trails, paths, closed walks, circuits, simple circuits, or just walks.