In Exercises 18-23 determine whether the directed graph...

In Exercises $18-23$ determine whether the directed graph shown has an Euler circuit. Construct an Euler circuit if one exists. If no Euler circuit exists, determine whether the directed graph has an Euler path. Construct an Euler path if one exists.

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Key Concepts

Euler Circuit

An Euler circuit is a closed trail in a graph that visits every edge exactly once and starts and ends at the same vertex. In the context of directed graphs, an Euler circuit exists if and only if the graph is strongly connected and every vertex has equal in-degree and out-degree.

Euler Path

An Euler path is a trail that visits every edge exactly once but does not necessarily start and end at the same vertex. For a directed graph, an Euler path exists if and only if the graph is connected and at most one vertex has an out-degree that exceeds its in-degree by one, at most one vertex has an in-degree that exceeds its out-degree by one, and all other vertices have equal in-degree and out-degree.

Directed Graph

A directed graph consists of a set of vertices connected by edges where each edge has a specified direction from one vertex to another. This directionality introduces constraints on the in-degrees and out-degrees of the vertices, which are critical for determining the existence of Euler paths and Euler circuits.

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