A chordal network is a data structure that represents polynomial ideals in terms of the paths of a certain directed graph. Remarkably, several polynomial ideals with exponentially many components admit compact chordal network representations. Moreover, chordal networks can be efficiently postprocessed to compute several properties of the underlying variety, such as cardinality, dimension, components, elimination ideals, and radical ideal membership.
We now present some examples.
Ideal of adjacent minors: Consider the ideal of adjacent minors of a $2\times n$ matrix . This ideal decomposes into $F_n$ components, where $F_n$ is the Fibonacci number. These (exponentially many) components can be represented compactly with a chordal network.





Once we have the chordal network, one may verify that the variety has codimension 9, and that it has $F_{10}=55$ components.

Edge ideals: The edge ideal of a graph $G=(V,E)$ is generated by the monomials $x_ix_j$ for $ij\in E$. Edge ideals have a very nice combinatorial structure, but they often have an exponential number of components. Chordal networks might be used to efficiently describe these large decompositions. The following code computes a chordal network representation for edge ideal of the product graph $C_3\times P_n$.






This variety has codimension 10, and has $48=3\times 2^{51}$ components.

Chromatic ideal of a cycle: Coloring graphs is a classical NPhard problem, but it is tractable for certain families of graphs. In particular, coloring the cycle graph $C_n$ is trivial. However, solving the problem algebraically (see chromaticIdeal) can be quite challenging using Gr\"obner bases. On the other hand, this chromatic ideal has a chordal network representation with less than $3n$ nodes [CP2017]. This network can be found with the command chordalTria(N), but the calculation requires Maple (see TriangularDecompAlgorithm).