Edge Intersection Research Articles

Barriles de absorción y contención del impacto: reducción de mortalidad por accidentes de tránsito

To determine the effect of the installation of an impact absorption and containment device (Spanish abbreviation, BAFI) on the reduction of traffic accident fatalities at edge crossings in the city of Monterrey, Mexico. Information about accidents was obtained from January 1,2000 to December 31,2005 from intersections with and without BAFI. During the study period, 142 accidents were registered in edge intersections, with 59 deaths in intersections without BAFIs and five in those where the devices were installed. The BAFI devices represent a quick and low-cost step for reducing mortality and serious injury due to accidents in edge crossings.

The [formula omitted]-edge intersection graphs of paths in a tree

We consider a generalization of edge intersection graphs of paths in a tree. Let P be a collection of nontrivial simple paths in a tree T . We define the k -edge ( k ⩾ 1 ) intersection graph Γ k ( P ) , whose vertices correspond to the members of P , and two vertices are joined by an edge if the corresponding members of P share k edges in T . An undirected graph G is called a k -edge intersection graph of paths in a tree, and denoted by k -EPT, if G = Γ k ( P ) for some P and T . It is known that the recognition and the coloring of the 1-EPT graphs are NP-complete. We extend this result and prove that the recognition and the coloring of the k -EPT graphs are NP-complete for any fixed k ⩾ 1 . We show that the problem of finding the largest clique on k -EPT graphs is polynomial, as was the case for 1-EPT graphs, and determine that there are at most O ( n 3 ) maximal cliques in a k -EPT graph on n vertices. We prove that the family of 1-EPT graphs is contained in, but is not equal to, the family of k -EPT graphs for any fixed k ⩾ 2 . We also investigate the hierarchical relationships between related classes of graphs, and present an infinite family of graphs that are not k -EPT graphs for every k ⩾ 2 . The edge intersection graphs are used in network applications. Scheduling undirected calls in a tree is equivalent to coloring an edge intersection graph of paths in a tree. Also assigning wavelengths to virtual connections in an optical network is equivalent to coloring an edge intersection graph of paths in a tree.

Representing edge intersection graphs of paths on degree 4 trees

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P , denoted by EPT ( P ) , has vertex set that corresponds to the members of P , and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G = EPT ( P ) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement G ¯ possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.

On the strong p-Helly property

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8–22]. Independently, other authors [A. Bretto, S. Ubéda, J. Žerovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55–57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216–220, W.D. Wallis, Guo-Hui Zhang, On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187–193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p = 2 , this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216–220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.

A global parallel algorithm for the hypergraph transversal problem

We consider the problem of finding all minimal transversals of a hypergraph H ⊆ 2 V , given by an explicit list of its hyperedges. We give a new decomposition technique for solving the problem with the following advantages: (i) Global parallelism: for certain classes of hypergraphs, e.g., hypergraphs of bounded edge size, and any given integer k, the algorithm outputs k minimal transversals of H in time bounded by polylog ( | V | , | H | , k ) assuming poly ( | V | , | H | , k ) number of processors. Except for the case of graphs, none of the previously known algorithms for solving the same problem exhibit this feature. (ii) Using this technique, we also obtain new results on the complexity of generating minimal transversals for new classes of hypergraphs, namely hypergraphs of bounded dual-conformality, and hypergraphs in which every edge intersects every minimal transversal in a bounded number of vertices.

Odd Crossing Number and Crossing Number Are Not the Same

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems).

The cost chromatic number and hypergraph parameters

In a graph, by deflnition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representations by edge intersections of hypergraphs.

Edge intersection graphs of linear 3-uniform hypergraphs

Abstract Let L 3 l be the class of edge intersection graphs of linear 3-uniform hypergraphs. The problem of recognizing G ∈ L 3 l is NP-complete. Denote by δ ALG the minimal integer such that the problem G ∈ L 3 l is polynomially solvable in the class of graphs G with the minimal vertex degree δ ( G ) ≥ δ ALG and by δ FIS the minimal integer such that L 3 l can be characterized by a finite list of forbidden induced subgraphs in the class of graphs G with δ ( G ) ≥ δ FIS . It is proved that δ ALG ≤ 10 and δ FIS ≤ 16 .

Disjoint representability of sets and their complements

For a hypergraph H and a set S, the trace of H on S is the set of all intersections of edges of H with S. We will consider forbidden trace problems, in which we want to find the largest hypergraph H that does not contain some list of forbidden configurations as traces, possibly with some restriction on the number of vertices or the size of the edges in H. In this paper we will focus on combinations of three forbidden configurations: the k-singleton [k](1), the k-co-singleton [k](k-1) and the k-chain Ck={∅,{1},[1,2],…,[1,k-1]}, where we write [k](ℓ) for the set of all ℓ-subsets of [k]={1,…,k}. Our main topic is hypergraphs with no k-singleton or k-co-singleton trace. We obtain an exact result in the case k=3, both for uniform and non-uniform hypergraphs, and classify the extremal examples. In the general case, we show that the number of edges in the largest r-uniform hypergraph with no k-singleton or k-co-singleton trace is of order rk-2. By contrast, Frankl and Pach showed that the number of edges in the largest r-uniform hypergraph with no k-singleton trace is of order rk-1. We also give a very short proof of the recent result of Balogh and Bollobás that there is a finite bound on the number of sets in any hypergraph without a k-singleton, k-co-singleton or k-chain trace, independently of the number of vertices or the size of the edges.

Representations of Edge Intersection Graphs of Paths in a Tree

Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

Strong branchwidth and local transversals

We consider representations of chordal graphs as edge intersection graphs of subtrees in a tree of maximum vertex degree 3. A new graph invariant related to the concept of branchwidth is introduced, and a structural characterization theorem is given for the existence of representations where each edge is contained in fewer subtrees than the clique number of the graph represented. As a consequence, a sufficient condition is obtained for chordal graphs to have smaller branchwidth than clique number.

Shear wave attenuation and dispersion in melt‐bearing olivine polycrystals: 2. Microstructural interpretation and seismological implications

The torsional forced oscillation tests of melt‐bearing olivine aggregates reported by Jackson et al. [2004] consistently show a peak in attenuation that is absent from melt‐free aggregates tested under similar conditions and grain sizes. Characterization by SEM shows that the melt resides in triple junction tubules and larger pockets as previously described. TEM imaging and EDS analysis reveals that olivine‐olivine grain boundaries are characterized by a region ≤1 nm wide which is structurally and chemically distinct from olivine grain interiors. From the possible mechanisms that can produce an anelastic attenuation peak, melt squirt can be eliminated for our samples and experimental conditions. We attribute the observed attenuation peak to elastically accommodated grain boundary sliding, requiring that the grain boundaries are weak relative to olivine grain interiors but have a significantly higher viscosity than bulk melt. While the nanometer scale grain boundary structure in the melt‐bearing aggregates is essentially the same as for melt‐free aggregates studied previously, elastically accommodated sliding in the latter is apparently inhibited by tight three‐grain edge intersections. The exponentially increasing high temperature background attenuation in both types of aggregate is attributed to diffusionally accommodated grain boundary sliding. Extrapolation to mantle grain sizes shows that the broad peak may be responsible for nearly frequency independent attenuation in partially molten regions of the upper mantle.

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.

Reducing feasible contacts between polyhedral models to red–blue intersections on the sphere

Orientation-related problems in geometric design can be naturally expressed on the spherical surface S 2. A wide subset of such problems can be solved directly on the sphere by adapting well-known planar data structures and algorithms. This paper shows that the detection of feasible contacts between two translating polyhedral models can be formulated as a problem of this type. First, a dual spherical representation of polyhedra is introduced, which reduces the contact detection above to finding intersections between two sets of spherical polygons. Next, the red–blue blocks plane sweep algorithm is adapted to obtain both edge intersections and point-in-polygon inclusions in the spherical setting. An experimental comparison of this algorithm against a naı̈ve one shows an increasing advantage of the former as the complexity of the setting grows. The obtained edge–edge and vertex–face polyhedral contacts provide the relevant feature pairs to be tested for interference, leading to considerable savings in collision detection between polyhedral models, as shown in the experimental test performed.

Full-field singular stresses in a composite laminate weakened by a cylindrical cavity: theory and experiment

Analytical and experimental investigation of strain fields arising in composite laminates in the vicinity of an open hole and ply interfaces was performed. The singular stress fields, arising at the ply interface and hole edge intersections, were analytically examined by using a novel technique based on superposition of an asymptotic and spline approximation solution. The moiré interferometry technique, providing high spatial resolution and allowing rapid strain variations across a laminate thickness to be recorded, was utilized to experimentally measure the strain distributions throughout the thickness of a [+302/-302/904]3s laminate. The singular solution and the experimentally measured strain showed good agreement for most strain components with some discrepancy for the interlaminar normal strain component.

On the stability number of the edge intersection of two graphs

Let G be the graph obtained as the edge intersection of two graphs G 1, G 2 on the same vertex set V. We show that if at least one of G 1, G 2 is perfect, then α( G)⩽ α( G 1)· α( G 2), where α() is the cardinality of the largest stable set. Moreover, for general G 1 and G 2, we show that α( G)⩽ R( α( G 1)+1, α( G 2)+1)−1, where R( k,ℓ) is the Ramsey number.

Visual motion of missing-fundamental patterns: motion energy versus feature correspondence

Missing-fundamental gratings, generated by subtracting the fundamental Fourier components from square-wave gratings, appear to move backward when presented in quarter-cycle jumps, even though their edges and features all move forward. We used variants of these stimuli to test current models of motion perception. We found that missing-fundamental plaids, constructed from orthogonal missing-fundamental gratings, also appear to move backward. Forward motion was restored to missing-fundamental gratings and plaids by adding back small fractions of the original fundamental. In-phase and antiphase addition of the fundamental had similar effects on the perceived motion, despite having markedly different effects on the features, appearances and zero-crossings of the stimuli. The critical amplitude of fundamental needed to restore forward motion to plaids was the same as that needed to restore forward motion to their isolated component gratings, indicating that the plaids’ emergent features, such as edge intersections and ‘blobs’, made little or no contribution to the perceived direction of motion in these stimuli. In two derivative experiments, missing-fundamental chromatic gratings and plaids, at approximate isoluminance, and missing-fundamental luminance barberpoles, also generated backward perceived motions, and these were also reversed by in-phase or antiphase addition of small amounts of fundamental.

Drawing Nice Projections of Objects in Space

Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three-dimensional Euclidean space, we consider the problem of computing a variety of “nice” parallel (orthographic) projections of the object. We show that given a general polygonal object consisting ofnline segments in space, deciding whether it admits acrossing-freeprojection can be done inO(n2logn+k) time andO(n2+k) space, wherekis the number of edge intersections of forbidden quadrilaterals (i.e., a set of directions that admits a crossing) and varies from zero toO(n4). This implies for example that, given a simple polygon in 3-space, we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, aminimum-crossingprojection can be found inO(n4) time and space. We show that an object always admits a regular projection (of interest to knot theory) and that such a projection can be obtained inO(n2) time and space or inO(n3) time and linear space. A description of the set of all directions which yield regular projections can be computed inO(n3logn+k) time, wherekis the number of intersections of a set of quadratic arcs on the direction sphere and varies fromO(n3) toO(n6). Finally, when the objects are polygons and trees in space, we considermonotonicprojections, i.e., projections such that every path from the root of the tree to every leaf is monotonic in a common direction on the projection plane. We solve a variety of such problems. For example, given a polygonal chainP, we can determine inO(n) time ifPis monotonic on the projection plane, and inO(nlogn) time we can findallthe viewing directions with respect to whichPis monotonic. In addition, inO(n2) time, we can determine all directions with respect to which a given tree or simple polygon is monotonic.

Nibble meshing: incremental triangulation of non-manifold solid boundary

A computational method, called Nibble algorithm, for triangulation of non-manifold solid boundary is proposed. The algorithm is based on an incremental boundary traversal technique. The mesh generator creates a mesh element-by-element until the whole region is covered no matter the domain complexity (faces with non-convex shapes and multi-connected boundaries are treated). At each step of the algorithm, a surface boundary called active boundary is evaluated in such a way that it nibbles the surface to be triangulated. The fundamental feature of this process is the definition of an area, called influence zone, which controls the node insertion and thus avoids edge intersection tests. Further, the generated mesh is refined through an extension of the Laplacian smoothing. It allows an optimization of the smoothing quality without saturating the time complexity. A new technique for adaptive smoothing is also applied in order to speed up the mesh refinement.

AnO(n 3) recognition algorithm for bithreshold graphs

A bithreshold graph is the edge intersection of two threshold graphs such that every independent set is independent in at least one of the threshold components. Recognizing a bithreshold graph is polynomially equivalent to recognizing its complement, i.e., a cobithreshold graph. In this paper we introduce a coloring of the vertices and of the edges of a cobithreshold graph that leads to a recognition and decomposition algorithm. This algorithm works inO(n 3) time improving the previously knownO(n 4) result [HM].