It was known that planar graphs have O(n) subgraphs isomorphic to K3 or K4. That is, K3 and K4 have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.
More generally, let F be a minor-closed family, and let x be the smallest number such that some complete bipartite graph Kx,y is a forbidden minor for F. Then the x-connected graphs have linear subgraph multiplicity for F, and there exists an (x − 1)-connected graph (namely Kx − 1,x − 1) that does not have linear subgraph multiplicity. When x ≤ 3 or when x = 4 and the minimal forbidden minors for F are triangle-free, then the graphs with linear subgraph multiplicity for F are exactly the x-connected graphs.
Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.
Shows that the minimum area polygon containing k of n points must be near a line determined by two points, and uses this observation to find the polygon quickly. Merged with "Iterated nearest neighbors and finding minimal polytopes" in the journal version.
We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness > t.
We show that thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness > t. The proof uses Ramsey-theoretic arguments similar to those in "Separating book thickness from thickness".
Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
Semi-automatically filtered from a common source file.