Mixed graph

In graph theory, a mixed graph is a graph consisting of a set of vertices, a set of edges, and a set of directed edges .

Definitions and notation

Consider adjacent vertices. A directed edge, called an arc, is an edge with an orientation and can be denoted as or . Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as or.
For the purpose of our example we will not be considering loops or multiple edges of mixed graphs.
A walk in a mixed graph is a sequence of vertices and edges/arcs such that for every index, either is an edge of the graph or is an arc of the graph. This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A walk is closed if its first and last vertices are the same, and a closed path is a cycle. A mixed graph is acyclic if it does not contain a cycle.

Coloring

Mixed graph coloring can be thought of as labeling or an assignment of different colors to the vertices of a mixed graph. Different colors must be assigned to vertices that are connected by an edge. The colors may be represented by the numbers from to, and for a directed arc, the tail of the arc must be colored by a smaller number than the head of the arc.

Example

For example, consider the figure to the right. Our available -colors to color our mixed graph are Since and are connected by an edge, they must receive different colors or labelings. We also have an arc from to. Since orientation assigns an ordering, we must label the tail with a smaller color than the head of our arc.

Strong and weak coloring

A proper -coloring of a mixed graph is a function where such that if and if.
A weaker condition on our arcs can be applied and we can consider a weak proper -coloring of a mixed graph to be a function where such that if and if.
Referring back to our example, this means that we can label both the head and tail of with the positive integer 2.

Counting

A coloring may or may not exist for a mixed graph. In order for a mixed graph to have a -coloring, the graph cannot contain any directed cycles. If such a -coloring exists, then we refer to the smallest needed in order to properly color our graph as the chromatic number, denoted by. The number of proper -colorings is a polynomial function of called the chromatic polynomial of our graph and can be denoted by.

Computing weak chromatic polynomials

The deletion–contraction method can be used to compute weak chromatic polynomials of mixed graphs. This method involves deleting an edge or arc and possibly joining the remaining vertices incident to that edge or arc to form one vertex. After deleting an edge from a mixed graph we obtain the mixed graph. We denote this deletion of the edge by. Similarly, by deleting an arc from a mixed graph, we obtain where we denote the deletion of by. Also, we denote the contraction of and by and, respectively. From Propositions given in Beck et al. we obtain the following equations to compute the chromatic polynomial of a mixed graph:

Applications

Scheduling problem

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. In this sort of problem, undirected edges may be used to model a constraint that two tasks are incompatible. Directed edges may be used to model precedence constraints, in which one task must be performed before another. A graph defined in this way from a scheduling problem is called a disjunctive graph. The mixed graph coloring problem can be used to find a schedule of minimum length for performing all the tasks.

Bayesian inference

Mixed graphs are also used as graphical models for Bayesian inference. In this context, an acyclic mixed graph is also called a chain graph. The directed edges of these graphs are used to indicate a causal connection between two events, in which the outcome of the first event influences the probability of the second event. Undirected edges, instead, indicate a non-causal correlation between two events. A connected component of the undirected subgraph of a chain graph is called a chain. 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.

Street network

A system of streets or roads may be represented by interconnecting mixed edges and nodes, where directed edges represent one-way streets and undirected edges represent two-way streets.