The coarseness of a graph is the maximum number of edge-disjoint nonplanar subgraphs contained in a given graph . The coarseness of a planar graph is therefore .
The coarseness of a graph is the sum of the coarsenesses of its blocks (Beineke and Chartrand 1968).
The coarseness of the complete graph is known for most values of except , divisible by 3 and greater than or equal to 18, and of the form . For all of these, the values are known to within 1 (Guy and Beineke 1968; Harary 1994, pp. 121-122).
The coarseness of the complete bipartite graph is known for values of satisfying certain conditions (Beineke and Guy 1969; Harary 1994, pp. 121-122).