Mixed graph

In graph theory, a mixed graph Template:Math is a graph consisting of a set of vertices Template:Mvar, a set of (undirected) edges Template:Mvar, and a set of directed edges (or arcs) Template:Mvar.

Definitions and notation

Example of a mixed graph

Consider adjacent vertices $u,v\in V$. A directed edge, called an arc, is an edge with an orientation and can be denoted as $\overrightarrow{uv}$ or $(u,v)$ (note that $u$ is the tail and $v$ is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as $uv$ or $[u,v]$.

Template:Further Template:Further 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 $v_0,c_1,v_1,c_2,v_2,\dots ,c_k,v_k$ of vertices and edges/arcs such that for every index $i$, either $c_i=v_i{v}_{i+1}$ is an edge of the graph or $c_i=\overrightarrow{v_i{v}_{i+1}}$ 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

Example of a mixed graph

Mixed graph coloring can be thought of as labeling or an assignment of Template:Mvar different colors (where Template:Mvar is a positive integer) 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 Template:Math to Template:Mvar, 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 Template:Mvar-colors to color our mixed graph are Template:Math Since Template:Mvar and Template:Mvar are connected by an edge, they must receive different colors or labelings (Template:Mvar and Template:Mvar are labelled 1 and 2, respectively). We also have an arc from Template:Mvar to Template:Mvar. Since orientation assigns an ordering, we must label the tail (Template:Mvar) with a smaller color (or integer from our set) than the head (Template:Mvar) of our arc.

Strong and weak coloring

A (strong) proper Template:Mvar-coloring of a mixed graph is a function Template:Math where Template:Math such that Template:Math if Template:Math and Template:Math if $\overrightarrow{uv}\in A$.

A weaker condition on our arcs can be applied and we can consider a weak proper Template:Mvar-coloring of a mixed graph to be a function Template:Math where Template:Math such that Template:Math if Template:Math and Template:Math if $\overrightarrow{uv}\in A$. Referring back to our example, this means that we can label both the head and tail of Template:Math 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 Template:Mvar-coloring, the graph cannot contain any directed cycles. If such a Template:Mvar-coloring exists, then we refer to the smallest Template:Mvar needed in order to properly color our graph as the chromatic number, denoted by Template:Math. The number of proper Template:Mvar-colorings is a polynomial function of Template:Mvar called the chromatic polynomial of our graph Template:Mvar (by analogy with the chromatic polynomial of undirected graphs) and can be denoted by Template:Math.

Computing weak chromatic polynomials

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

1. ${\chi }_G(k)={\chi }_{G-e}(k)-{\chi }_{G/e}(k)$,
2. ${\chi }_G(k)={\chi }_{G-a}(k)+{\chi }_{G/a}(k)-{\chi }_{G_a}(k)$.

Applications

Scheduling problem

Template:Main 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 (they cannot be performed simultaneously). 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 (one with no cycles of directed edges) 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.Template:Sfnp

References

• Template:Citation.
• Template:Citation
• Template:Citation.
• Template:Citation.

External links

• Template:Mathworld