Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈?, a $\frac{1}{r}We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a \frac1r\frac{1}{r} -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ³ and a parameter r∈ℕ, we construct a \frac1r\frac{1}{r} -cutting of size O(r ²). For n axis-aligned rectangles in ℝ³, we construct a \frac1r\frac{1}{r} -cutting of size O(r ^3/2). As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in ℝ³, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O(?n)O(\sqrt{n}) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n ^1/3) edges. Both bounds are best possible.     

Spanning Trees Crossing Few Barriers

We consider the problem of finding low-cost spanning trees for sets of $n$ points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of $m$ barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost $O(\min(m^2,m\sqrt{n}))$; (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost $O(m)$; (iii) ] if the barriers are disjoint convex objects, then there is always a spanning tree of cost $O(n+m)$. All our bounds are worst-case optimal, up to multiplicative constants.     

Disjoint Compatible Geometric Matchings

We prove that for every set of n pairwise disjoint line segments in the plane in general position, where n is even, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. (Comput. Geom. 42:617–626, 2009). The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.     

Multicolored trees in complete graphs

A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K_2n(n > 2) with 2n ? 1 colors, there are two edge‐disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a₁,…, a_k) is a color distribution for the complete graph K_n, n ≥ 5, such that , then there exist two edge‐disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph K_n with the above distribution if T is a non‐star multicolored spanning tree of K_n, then there exists a multicolored spanning tree T' of K_n such that T and T' are edge‐disjoint. Also it is shown that if K_n, n ≥ 6, is edge colored with k colors and , then there exist two edge‐disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221–232, 2007     

Pointed binary encompassing trees: Simple and optimal

For n disjoint line segments in the plane we construct in optimal O(nlogn) time and linear space an encompassing tree of maximum degree three such that at every vertex v all edges of the tree that are incident to v lie in a halfplane bounded by the line through the input segment which v is an endpoint of. In particular, this tree is pointed since every vertex has an incident angle greater than π. Such a pointed binary tree can be augmented to a minimum pseudo-triangulation. It follows that every set of disjoint line segments in the plane has a constrained minimum pseudo-triangulation whose maximum vertex degree is bounded by a constant.     

Every Set of Disjoint Line Segments Admits a Binary Tree

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires degree 3. We provide an O(nlog n) time algorithm for constructing such a tree, and show that this is optimal. Received September 14, 1999, and in revised form January 17, 2001. Online publication August 29, 2001.     

Geometric Graphs with Few Disjoint Edges

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and T?rőcsik, we show that a geometric graph on n vertices with no k+1 pairwise disjoint edges has at most k ³ (n+1) edges. On the other hand, we construct geometric graphs with n vertices and approximately (3/2)(k-1)n edges, containing no k+1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski, and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges. Received May 7, 1998, and in revised form March 24, 1999.     

On Characterizations of Rigid Graphs in the Plane Using Spanning Trees

We study characterizations of generic rigid graphs and generic circuits in the plane using only few decompositions into spanning trees. Generic rigid graphs in the plane can be characterized by spanning tree decompositions [5,6]. A graph G with n vertices and 2n − 3 edges is generic rigid in the plane if and only if doubling any edge results in a graph which is the union of two spanning trees. This requires 2n − 3 decompositions into spanning trees. We show that n − 2 decompositions suffice: only edges of G − T can be doubled where T is a spanning tree of G. A recent result on tensegrity frameworks by Recski [7] implies a characterization of generic circuits in the plane. A graph G with n vertices and 2n − 2 edges is a generic circuit in the plane if and only if replacing any edge of G by any (possibly new) edge results in a graph which is the union of two spanning trees. This requires decompositions into spanning trees. We show that 2n − 2 decompositions suffice. Let be any circular order of edges of G (i.e. ). The graph G is a generic circuit in the plane if and only if is the union of two spanning trees for any . Furthermore, we show that only n decompositions into spanning trees suffice.     

A probably fast,provably optimal algorithm for rectilinear Steiner trees

We use the technique of divide-and-conquer to construct a rectilinear Steiner minimal tree on a set of sites in the plane. A well-known optimal algorithm for this problem by Dreyfus and Wagner [10] is used to solve the problem in the base case. The run time of our optimal algorithm is probabilistic in nature: for all ? > 0, there exists b > 0 such that Prob[T(n) > 2^{b√n log n]>1–?, for n log n} > 1 – ?, for n sites uniformly distributed on a rectangle. The key fact in the run-time argument is the existence of probable bounds on the number of edges of an optimal tree crossing our subdivision lines. We can test these bounds in low-degree polynomial time for any given set of sites. © 1994 John Wiley & Sons, Inc.     

Über eine eigenschaft lokalfiniter,unendlicher bäume

Let A be an arbitrary locally finite, infinite tree and assume that a graph G contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. This sharpens a theorem of Halin [5], who proved the corresponding result for the case that A is a tree in which each vertex has degree not greater than 3.     

Network reinforcement

We give an algorithm for the following problem: given a graph G=(V,E) with edge-weights and a nonnegative integer k, find a minimum cost set of edges that contains k disjoint spanning trees. This also solves the following reinforcement problem: given a network, a number k and a set of candidate edges, each of them with an associated cost, find a minimum cost set of candidate edges to be added to the network so it contains k disjoint spanning trees. The number k is seen as a measure of the invulnerability of a network. Our algorithm has the same asymptotic complexity as |V| applications of the minimum cut algorithm of Goldberg & Tarjan. Received: April, 2004     

Subdivisions of graphs: A generalization of paths and cycles

One of the basic results in graph theory is Dirac's theorem, that every graph of order n?3 and minimum degree ?n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree ?n/2 contains a cycle C then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n-t)/2 if H has t components that are trees.We attempt a similar generalization of the Corrádi-Hajnal theorem that every graph of order ?3k and minimum degree ?2k contains k disjoint cycles. Again, this may be restated as: every graph of order ?3k and minimum degree ?2k contains a subdivision of kK₃. We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order ?n and minimum degree ?n-k contains a subdivision of H.     

The spanning trees forced by the path and the star

A set S of trees of order n forces a tree T if every graph having each tree in S as a spanning tree must also have T as a spanning tree. A spanning tree forcing set for order n that forces every tree of order n. A spanning-tree forcing set S is a test set for panarboreal graphs, since a graph of order n is panarboreal if and only if it has all of the trees in S as spanning trees. For each positive integer n ≠ 1, the star belongs to every spanning tree forcing set for order n. The main results of this paper are a proof that the path belongs to every spanning-tree forcing set for each order n ∉ {1, 6, 7, 8} and a computationally tractable characterization of the trees of order n ≥ 15 forced by the path and the star. Corollaries of those results include a construction of many trees that do not belong to any minimal spanning tree forcing set for orders n ≥ 15 and a proof that the following related decision problem is NP-complete: an instance is a pair (G, T) consisting of a graph G of order n and maximum degree n - 1 with a hamiltonian path, and a tree T of order n; the problem is to determine whether T is a spanning tree of G. © 1996 John Wiley & Sons, Inc.     

Graphs With 3n−6 Edges Not Containing A Subdivision Of K 5

We determine all graphs on n ≥ 3 vertices with 3n-6 edges which do not contain a subdivision of K₅. These are exactly the graphs which one gets from any number of disjoint maximal planar graphs by successively pasting along triangles.     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

   Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

Every DFS Tree of a 3‐Connected Graph Contains a Contractible Edge

Tutte proved that every 3‐connected graph G on more than 4 vertices contains a contractible edge. We strengthen this result by showing that every depth‐first‐search tree of G contains a contractible edge. Moreover, we show that every spanning tree of G contains a contractible edge if G is 3‐regular or if G does not contain two disjoint pairs of adjacent degree‐3 vertices.     

Degree Condition for Subdivisions of Unicyclic Graphs

If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.     

Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ³ in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ ^d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.