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Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with n vertices has a F~ry embedding (i.e., straight-line embedding) on the 2n-4 by n 2 grid and provide an O(n) space, O(n log n) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a(More)
A theory of cognitive mapping is developed that depends only on accepted properties of hippocampal function, namely, long-term potentiation, the place cell phenomenon, and the associative or recurrent connections made among CA3 pyramidal cells. It is proposed that the distance between the firing fields of connected pairs of CA3 place cells is encoded as(More)
Given a simple graph G, let v(G) and e(G) denote its number of vertices and edges, respectively. We say that G is drawn in the plane if its vertices are represented by distinct points of the plane and its edges are represented by Jordan arcs connecting the corresponding point pairs but not passing through any other vertex. Throughout this paper, we only(More)
A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number)(More)
Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with <italic>n</italic> vertices has a F&#225;ry embedding (i.e., straight-line embedding) on the 2<italic>n</italic> - 4 by <italic>n</italic> - 2 grid and provide an &Ogr;(<italic>n</italic>) space, &Ogr;(<italic>n</italic> log <italic>n</italic>) time algorithm to effect this(More)