YΔ- and ΔY-transformation

In graph theory, ΔY- and YΔ-transformations (also written delta-wye and wye-delta) are a pair of operations on graphs.

A ΔY-transformation replaces a triangle by a vertex of degree three; and conversely, a YΔ-transformation replaces a vertex of degree three by a triangle.

The names for the operations derive from the shapes of the involved subgraphs, which look respectively like the letter Y and the Greek capital letter Δ.

A YΔ-transformation may create parallel edges, even if applied to a simple graph.

For this reason ΔY- and YΔ-transformations are most naturally considered as operations on multigraphs.

On multigraphs both operations preserve the edge count and are exact inverses of each other.

In the context of simple graphs it is common to combine a YΔ-transformation with a subsequent normalization step that reduces parallel edges to a single edge.

This may no longer preserve the number of edges, nor be exactly reversible via a ΔY-transformation.

be a graph (potentially a multigraph).

is a vertex of degree three with neighbors

If the resulting graph should be a simple graph, then any resulting parallel edges are to be replaced by a single edge.

ΔY- and YΔ-transformations are a tool both in pure graph theory as well as applications.

Both operations preserve a number of natural topological properties of graphs.

For example, applying a YΔ-transformation to a 3-vertex of a planar graph, or a ΔY-transformation to a triangular face of a planar graph, results again in a planar graph.

[1] This was used in the original proof of Steinitz's theorem, showing that every 3-connected planar graph is the edge graph of a polyhedron.

Applying ΔY- and YΔ-transformations to a linkless graph results again in a linkless graph.

[2] This fact is used to compactly describe the forbidden minors of the associated graph classes as ΔY-families generated from a small number of graphs (see the section on ΔY-families below).

A particularly relevant application exists in electrical engineering in the study of three-phase power systems (see Y-Δ transform (electrical engineering)).

The ΔY-family generated by a graph

is the smallest family of graphs that contains

and is closed under YΔ- and ΔY-transformations.

by recursively applying these transformations until no new graph is generated.

is a finite graph it generates a finite ΔY-family, all members of which have the same edge count.

The ΔY-family generated by several graphs is the smallest family that contains all these graphs and is closed under YΔ- and ΔY-transformation.

Some notable families are generated in this way: A graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of ΔY- or YΔ-transformations and the following normalization steps: The YΔY-reducible graphs form a minor closed family and therefore have a forbidden minor characterization (by the Robertson–Seymour theorem).

The graphs of the Petersen family constitute some (but not all) of the excluded minors.

[5] In fact, already more than 68 billion excluded minor are known.