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A graph-encoded map (gray triangles and colored edges) of a graph in the plane (white circles and black edges)

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

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

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

References