A path is closed if its first and last vertices are the same, and a closed path is a cycle if it does not repeat vertices, except the first and the last. For example, consider the figure to the right. A chain graph may be transformed into an undirected graph by constructing its moral graph , an undirected graph formed from the chain graph by adding undirected edges between pairs of vertices that have outgoing edges to the same chain, and then forgetting the orientations of the directed edges. In order for a mixed graph to have a k-coloring, the graph cannot contain any directed cycles. The mixed graph coloring problem can be used to find a schedule of minimum length for performing all the tasks. A mixed graph is acyclic if it does not contain a cycle. This method involves deleting or removing an edge or arc and contracting or joining the remaining vertices incident to that edge or arc to form one vertex. For the purpose of our application example we will not be considering loops or multiple edges of mixed graphs. Mixed graphs may be used to model job shop scheduling problems in which a collection of tasks is to be performed, subject to certain timing constraints.
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