Geometric biplane graphs I: maximal graphs

We study biplane graphs drawn on a finite planar point set in general position. This is the family of geometric graphs whose vertex set is and can be decomposed into two plane graphs. We show that two maximal biplane graphs-in the sense that no edge can be added while staying biplane-may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over -element point sets.Peer ReviewedPostprint (author's final draft

Geometric biplane graphs II: graph augmentation

We study biplane graphs drawn on a finite point set in the plane in general position. This is the family of geometric graphs whose vertex set is and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author's final draft

Geometric biplane graphs I: maximal graphs

Postprint (author’s final draft

Blocking the k-holes of point sets in the plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5Peer ReviewedPostprint (author's final draft

Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a nite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every su ciently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6- connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author’s final draft

Total domination in plane triangulations

Preprin

Compatible spanning trees

Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree of S. In this work, we study the problem of finding a second plane geometric tree T' spanning S, such that is compatible with T and shares the minimum number of edges with T. We prove that there is always a compatible plane geometric tree T' having at most #n - 3#/4 edges in common with T, and that for some plane geometric trees T, any plane tree T' spanning S, compatible with T, has at least #n - 2#/5 edges in common with T. #C# 2013 Elsevier B.V. All rights reserved.Preprin

Compatible spanning trees

Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree of S. In this work, we study the problem of finding a second plane geometric tree T' spanning S, such that is compatible with T and shares the minimum number of edges with T. We prove that there is always a compatible plane geometric tree T' having at most #n - 3#/4 edges in common with T, and that for some plane geometric trees T, any plane tree T' spanning S, compatible with T, has at least #n - 2#/5 edges in common with T. #C# 2013 Elsevier B.V. All rights reserved

Configurations of non-crossing rays and related problems

Let S be a set of n points in the plane and let R be a set of n pairwise non-crossing rays, each with an apex at a different point of S. Two sets of non-crossing rays R1R1 and R2R2 are considered to be different if the cyclic permutations they induce at infinity are different. In this paper, we study the number r(S) of different configurations of non-crossing rays that can be obtained from a given point set S. We define the extremal values r¯¯(n)=max|S|=nr(S) and r–(n)=min|S|=nr(S), r¯(n)=max|S|=nr(S) and r_(n)=min|S|=nr(S), and we prove that r–(n)=O*(2n)r_(n)=O*(2n) , r–(n)=O*(3.516n)r_(n)=O*(3.516n) and that r¯¯(n)=T*(4n)r¯(n)=T*(4n) . We also consider the number of different ways, r¿(S)r¿(S) , in which a point set S can be connected to a simple curve ¿¿ using a set of non-crossing straight-line segments. We define and study r¯¯¿(n)=max|S|=nr¿(S)and r–¿(n)=min|S|=nr¿(S), r¯¿(n)=max|S|=nr¿(S)and r_¿(n)=min|S|=nr¿(S), and we find these values for the following cases: When ¿¿ is a line and the points of S are in one of the halfplanes defined by ¿¿ , then r–¿(n)=T*(2n)r_¿(n)=T*(2n) and r¯¯¿(n)=T*(4n)r¯¿(n)=T*(4n) . When ¿¿ is a convex curve enclosing S, then r¯¯¿(n)=O*(16n)r¯¿(n)=O*(16n) . If all the points of S belong to a convex closed curve ¿¿ , then r–¿(n)=r¯¯¿(n)=T*(5n)r_¿(n)=r¯¿(n)=T*(5n) .Peer Reviewe

Blocking the k-holes of point sets in the plane

Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the k-holes of P if any k-hole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of k-hole blocking sets, with emphasis in the case k=5Peer Reviewe