Graph Thickness

The thickness (or depth) (Skiena 1990, p. 251; Beineke 1997) or (Harary 1994, p. 120) of a graph is the minimum number of planar edge-induced subgraphs of needed such that the graph union (Skiena 1990, p. 251).

The thickness of a planar graph is therefore , and the thickness of a nonplanar graph satisfies . A graph which is the union of two planar graph (i.e., that has thickness 1 or 2) is said to be a biplanar graph (Beineke 1997).

Determining the thickness of a graph is an NP-complete problem (Mansfeld 1983, Beineke 1997). Precomputed thicknesses for many small named or indexed graphs can be obtained in the Wolfram Language using GraphData[graph, "Thickness"].

A lower bound for the thickness of a graph is given by

(1)

where is the number of edges, is the number vertices, and is the ceiling function (Skiena 1990, p. 251). The example above shows a decomposition of the complete graph into three planar subgraphs. This decomposition is minimal, so , in agreement with the bound .

It follows from Brooks' theorem that the thickness of a graph is at most one more than the graph's local crossing number (Kainen 1973, Thomassen 1988).

The thickness of a complete graph satisfies

(2)

except for (Vasak 1976, Alekseev and Gonchakov 1976, Beineke 1997). For , 2, ..., the thicknesses are therefore 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, ... (OEIS A124156).

The thickness of a complete bipartite graph is given by

(3)

except possibly when and are both odd and, taking , there exists an even integer with (Beineke et al. 1964; Harary 1994, p. 121; Beineke 1997, where the ceiling in the exceptional condition given by Beineke 1997 has been corrected to a floor). The smallest such exceptional values are summarized in the following table.

13	17	4
17	21	5
19	29	6
19	47	7
21	25	6
23	75	9
25	29	7
25	59	9

According to Beineke (1997), the only subset of exceptional bipartite indices for are , , , , and .

The thickness of is therefore given by

(4)

(Harary 1994, p. 121), which for , 2, ... give the values 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, ... (OEIS A128929).

Finally, the thickness of a hypercube graph is given by

(5)

(Harary 1994, p. 121), which for , 2, ... give the values 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, (OEIS A144075).

A number of variations of graph thickness such as outerplanar thickness, arboricity, book thickness, and toroidal thickness have also been introduced (Beineke 1997).