Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing
نویسندگان
چکیده
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least 2(n−54)/4 nonisomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete. Running head: Obstructions for 1-immersions Corresponding author: AMS classifications: 05B07, 05C10.
منابع مشابه
Hardness of embedding simplicial complexes in R
Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding ofK into R? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3 (even if k is not considered fixed). We show that the celebrated result of Novikov on the...
متن کاملHardness of embedding simplicial complexes in Rd
Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into Rd? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of reco...
متن کاملOrdered Level Planarity and Geodesic Planarity
We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We show NP-completeness even for the spe...
متن کاملIntrinsic Obstructions to the Existence of Isometric Minimal Immersions
S.S. Chern raised the problem to find necessary and sufficient conditions for a given Riemannian manifold to be realizable on a minimal submanifold of a Euclidean space. The aim of this paper is to provide new necessary conditions. For minimal submanifolds in a Euclidean space we consider the negative of the Ricci tensor as defining a new metric, which is nothing but the third fundamental form,...
متن کاملAlmost All Friendly Matrices Have Many Obstructions
A symmetric m×m matrix M with entries taken from {0, 1, ∗} gives rise to a graph partition problem, asking whether a graph can be partitioned into m vertex sets matched to the rows (and corresponding columns) of M such that, if Mij = 1, then any two vertices between the corresponding vertex sets are joined by an edge, and if Mij = 0 then any two vertices between the corresponding vertex sets ar...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008