placementgenus

Placementgenus is a theoretical invariant discussed in the context of constrained graph embeddings. It measures how the requirement to place graph vertices at fixed locations on a surface affects the minimum genus needed to realize the drawing without edge crossings.

Formally, let G = (V,E) be a finite graph and let φ be a placement map that assigns to each vertex v in V a location φ(v) on a compact orientable surface S. The placementgenus pg(G, φ) is defined as the smallest genus g such that G can be embedded into the orientable surface S_g of genus g with each vertex v mapped to φ(v). If no fixed positions are imposed (that is, φ is unconstrained), then pg(G, φ) reduces to the ordinary genus g(G) of the graph.

The placementgenus is a refinement of the standard genus that reflects placement constraints. In general, pg(G, φ) ≥ g(G); fixed vertex locations can force embeddings on surfaces of higher genus or preclude embeddings that would be possible with unfixed placement. Relationships among placementgenus, crossing number, and other embedding parameters are active areas of study in topological graph theory.

Computationally, determining pg(G, φ) is typically difficult and can be NP-hard in various formulations, mirroring the complexity of related embedding and drawing problems. Research on placementgenus intersects graph drawing, network layout, and VLSI design, where fixed component positions arise naturally.

See also: graph genus, graph embedding, crossing number, graph drawing.