Assertion: graph | /r/HasProperty | planar
Facets:
Source sentences:
text: A graph is planar if it contains as a minor neither the complete bipartite graph K 3,3 (see the Three-cottage problem) nor the complete graph K 5 .
text: A graph is planar if it contains as a subdivision neither the complete bipartite graph K 3,3 nor the complete graph K 5 .
text: However, not every (3,6)-sparse graph is planar.
text: If the graph is planar, then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a graph embedding.
text: However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite graph K 3,3 as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions.
text: If the graph is planar and has low branch-width (or is nonplanar but with a known branch decomposition), then it can be solved in polynomial time using dynamic programming.
text: The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K 5 nor the complete bipartite graph K 3,3 .
text: All 1-vertex-connected cubic well-covered graphs have this form, and all such graphs are planar.
text: One of these two families is formed by replacing the nodes of a cycle by fragments A and B, with at least two of the fragments being of type A; a graph of this type is planar if and only if it does not contain any fragments of type B.
text: The other family is formed by replacing the nodes of a path by fragments of type B and C; all such graphs are planar.