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.^[1] It is the topological analogue of runcination, a geometric operation on polyhedra. Graph-encoded maps were formulated and named by Lins (1982).^[2] Alternative and equivalent systems for representing cellular embeddings include signed rotation systems and ribbon graphs.

The graph-encoded map for an embedded graph ${\displaystyle G}$ is another cubic graph ${\displaystyle H}$ together with a 3-edge-coloring of ${\displaystyle H}$. Each edge ${\displaystyle e}$ of ${\displaystyle G}$ is expanded into exactly four vertices in ${\displaystyle H}$, one for each choice of a side and endpoint of the edge. An edge in ${\displaystyle H}$ connects each such vertex to the vertex representing the opposite side and same endpoint of ${\displaystyle e}$; these edges are by convention colored red. Another edge in ${\displaystyle H}$ connects each vertex to the vertex representing the opposite endpoint and same side of ${\displaystyle e}$; these edges are by convention colored blue. An edge in ${\displaystyle H}$ of the third color, yellow, connects each vertex to the vertex representing another edge ${\displaystyle e'}$ that meets ${\displaystyle e}$ at the same side and endpoint.^[1]

An alternative description of ${\displaystyle H}$ is that it has a vertex for each flag of ${\displaystyle G}$ (a mutually incident triple of a vertex, edge, and face). If ${\displaystyle (v,e,f)}$ is a flag, then there is exactly one vertex ${\displaystyle v'}$, edge ${\displaystyle e'}$, and face ${\displaystyle f'}$ such that ${\displaystyle (v',e,f)}$, ${\displaystyle (v,e',f)}$, and ${\displaystyle (v,e,f')}$ are also flags. The three colors of edges in ${\displaystyle H}$ 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 ${\displaystyle (v,e,f)}$ may be associated with up to four different vertices of the gem.^[1]

Whenever a cubic graph ${\displaystyle H}$ 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 ${\displaystyle G}$. To recover ${\displaystyle G}$ and its embedding, interpret each 2-colored cycle of ${\displaystyle H}$ as the face of an embedding of ${\displaystyle H}$ onto a surface, contract each red--yellow cycle into a single vertex of ${\displaystyle G}$, and replace each pair of parallel blue edges left by the contraction with a single edge of ${\displaystyle G}$.^[1]

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.^[3]