Graph-encoded map

In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices per edge of the original graph. It is the topological analogue of runcination, a geometric operation on polyhedra. Graph-encoded maps were formulated and named by. Alternative and equivalent systems for representing cellular embeddings include signed rotation systems and ribbon graphs.

The graph-encoded map for an embedded graph $$G$$ is another cubic graph $$H$$ together with a 3-edge-coloring of $$H$$. Each edge $$e$$ of $$G$$ is expanded into exactly four vertices in $$H$$, one for each choice of a side and endpoint of the edge. An edge in $$H$$ connects each such vertex to the vertex representing the opposite side and same endpoint of $$e$$; these edges are by convention colored red. Another edge in $$H$$ connects each vertex to the vertex representing the opposite endpoint and same side of $$e$$; these edges are by convention colored blue. An edge in $$H$$ of the third color, yellow, connects each vertex to the vertex representing another edge $$e'$$ that meets $$e$$ at the same side and endpoint.

An alternative description of $$H$$ is that it has a vertex for each flag of $$G$$ (a mutually incident triple of a vertex, edge, and face). If $$(v,e,f)$$ is a flag, then there is exactly one vertex $$v'$$, edge $$e'$$, and face $$f'$$ such that $$(v',e,f)$$, $$(v,e',f)$$, and $$(v,e,f')$$ are also flags. The three colors of edges in $$H$$ represent each of these three types of flags that differ by one of their three elements. However, interpreting a graph-encoded map in this way requires more care. When the same face appears on both sides of an edge, as can happen for instance for a planar embedding of a tree, the two sides give rise to different gem vertices. And when the same vertex appears at both endpoints of a self-loop, the two ends of the edge again give rise to different gem vertices. In this way, each triple $$(v,e,f)$$ may be associated with up to four different vertices of the gem.

Whenever a cubic graph $$H$$ can be 3-edge-colored so that the red-blue cycles of the coloring all have length four, the colored graph can be interpreted as a graph-encoded map, and represents an embedding of another graph $$G$$. To recover $$G$$ and its embedding, interpret each 2-colored cycle of $$H$$ as the face of an embedding of $$H$$ onto a surface, contract each red--yellow cycle into a single vertex of $$G$$, and replace each pair of parallel blue edges left by the contraction with a single edge of $$G$$.

The dual graph of a graph-encoded map may be obtained from the map by recoloring it so that the red edges of the gem become blue and the blue edges become red.