Pair Of Arcs Research Articles

Almost disjoint paths and separating by forbidden pairs

By Menger's theorem, the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minimum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem. Moreover, the max-flow and min-cut problems form a strongly dual pair. We relax the disjointedness requirement on the paths, allowing them to be almost disjoint, meaning they may share up to one arc. The resulting almost disjoint paths problem (ADP) asks for k s-t-paths such that any two of them are almost disjoint. The separating by forbidden pairs problem (SFP) is the corresponding dual problem and calls for a set of k arc pairs such that every s-t-path contains both arcs of at least one such pair.In this paper, we explore these two problems, showing that they have an unbounded duality gap in general and analyzing their complexity. We prove that ADP is NP-complete when k is part of the input and that SFP is Σ2P-complete, even for acyclic graphs. Furthermore, we efficiently solve ADP when k≤2 is fixed and present a polynomial time algorithm based on dynamic programming for ADP when k is constant.

Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains

Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains

Shortest paths with exclusive-disjunction arc pairs conflicts

In this paper, a NP-hard variant of the shortest path problem, involving exclusive-disjunction arc pairs conflicts, is introduced. In this framework, a conflict is violated and a penalty has to be paid if either both the arcs in a pair or none of them are selected. The aim is to find a path for which the overall cost, defined as the sum of the costs of the traversed arcs and the penalties of the violated conflicts, is minimized. The proposed variant is used to model web applications test planning scenarios, where a path represents a sequence of web pages and hyperlinks. A proof of the NP-hardness is provided and two mathematical programming formulations are proposed for the problem. The former is an integer linear program relying on auxiliary variables to model conflict violations, while the latter is characterized by a nonlinear objective function, that uses only the arc variables to derive the penalties associated to the violated conflicts. To solve the problem, a two-stage matheuristic algorithm is also presented and its performances are compared with those provided by the best formulation solved by CPLEX.

Disproof of a conjecture on the rainbow triangles in arc-colored digraphs

Let D be a digraph. The arc number a(D) of D is defined as the number of arcs of D. An arc-coloring of D is a mapping C: A(D)→N, where N is the set of natural numbers. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Recently, Li et al. (2022) proved that if a(D)+c(D)=n(n−1)+⌊n24⌋+2 for an arc-colored digraph D of order n=3,4 without containing rainbow triangles, then D≅Kn↔. Thus, they conjectured that if a(D)+c(D)=n(n−1)+⌊n24⌋+1 for an arc-colored digraph D of order n≥5 without containing rainbow triangles, then D≅Kn↔. In this note, we disprove the conjecture by constructing a family of counterexamples. In fact, we show that there is not a function f(n) such that D≅Kn↔ if a(D)+c(D)=f(n) for an arc-colored digraph D of order n≥5 without containing rainbow triangles.

Distinguishing arc-colourings of symmetric digraphs

A symmetric digraph is obtained from an undirected graph by replacing each edge uv of by a pair of opposite arcs and . An arc-colouring of a digraph is called distinguishing if the only automorphism preserving it is the identity. The least number of colours in a distinguishing arc-colouring, not necessarily proper, of is called the distinguishing index D'(). We study bounds for D'(). For proper distinguishing arc-colourings, the least number of colours is called the distinguishing chromatic index of . There are 15 possible types of proper arc-colourings of a digraph depending on the definition of adjacent arcs. In this paper we investigate distinguishing chromatic indices of for the nine remaining types not considered in our two previous papers. We formulate several conjectures.

Rainbow triangles in arc-colored digraphs

Let D be an arc-colored digraph. The arc number a(D) of D is defined as the number of arcs of D. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Let f(D) be the smallest integer such that if c(D)≥f(D), then D contains a rainbow triangle. In this paper we obtain f(K↔n) and f(Tn), where K↔n is a complete digraph of order n and Tn is a strongly connected tournament of order n. Moreover we characterize the arc-colored complete digraph K↔n with c(K↔n)=f(K↔n)−1 and containing no rainbow triangles. We also prove that an arc-colored digraph D on n vertices contains a rainbow triangle when a(D)+c(D)≥a(K↔n)+f(K↔n), which is a directed extension of the undirected case.

Latest Triassic–Early Jurassic Stikine–Yukon-Tanana terrane collision and the onset of accretion in the Canadian Cordillera: Insights from Hazelton Group detrital zircon provenance and arc–back-arc configuration

AbstractThe Hazelton Group is a Rhaetian–Bajocian (uppermost Triassic–Middle Jurassic) volcano-sedimentary sequence that represents both the last pre-accretionary arc volcanic cycle of Stikinia and its early synaccretionary aftermath. Hazelton magmatism of central Stikinia succeeded the Late Triassic (mainly Carnian–Norian) Stuhini arc, which ceased activity as a result of end-on collision with the pericratonic Yukon-Tanana terrane. The Hazelton volcanic belt lies to the south along strike with the coeval Whitehorse trough, the synorogenic clastic basin that developed on top of the Stikinia–Yukon-Tanana collision zone. Whereas the sources of voluminous clastic sediments in the Whitehorse trough were its rapidly exhuming shoulders, the thin clastic intervals in the Hazelton Group in northwestern British Columbia were derived from local to subregional block uplifts that supplied mainly ca. 230–215 Ma zircons eroded from the plutonic roots of the Stuhini arc. Lesser components include late Paleozoic (ca. 350–330 Ma) zircons from Stikinia's basement and penecontemporaneous (ca. 205–172 Ma) zircons from Hazelton volcanic/subvolcanic sources. Reexamination of the four main volcanic fields that make up the lower Hazelton Group suggests that the main Hazelton volcanic belt formed a southward-convex magmatic arc from eastern Stikinia across the Skeena arch, including the Toodoggone and Telkwa belts, with the Spatsizi and Stewart-Iskut regions of northwestern British Columbia in its back-arc. The Whitehorse trough and Hazelton belt represent a collision zone to active arc pair. Southward advance of the arc and counterclockwise rotation of the Stikinia microplate contributed to closure against the Quesnellia arc and assembly of the inner Canadian Cordilleran terrane collage.

Canonical Embeddings of Pairs of Arcs

We show that for given four points in the Riemann sphere and a given isotopy class of two disjoint arcs connecting these points in two pairs, there exists a unique configuration with the property that each arc is a hyperbolic geodesic segment in the complement of the other arc.

Effective Organs-at-Risk Dose Sparing in Volumetric Modulated Arc Therapy Using a Half-Beam Technique in Whole Pelvic Irradiation.

BackgroundAlthough there are some controversies regarding whole pelvic radiation therapy (WPRT) due to its gastrointestinal and hematologic toxicities, it is considered for patients with gynecological, rectal, and prostate cancer. To effectively spare organs-at-risk (OAR) doses using multi-leaf collimator (MLC)’s optimal segments, potential dosimetric benefits in volumetric modulated arc therapy (VMAT) using a half-beam technique (HF) were investigated for WPRT.MethodsWhile the size of a fully opened field (FF) was decided to entirely include a planning target volume in all beam’s eye view across arc angles, the HF was designed to use half the FF from the isocenter for dose optimization. The left or the right half of the FF was alternatively opened in VMAT-HF using a pair of arcs rotating clockwise and counterclockwise. Dosimetric benefits of VMAT-HF, presented with dose conformity, homogeneity, and dose–volume parameters in terms of modulation complex score, were compared to VMAT optimized using the FF (VMAT-FF). Consequent normal tissue complication probability (NTCP) by reducing the irradiated volumes was evaluated as well as dose–volume parameters with statistical analysis for OAR. Moreover, beam-on time and MLC position precision were analyzed with log files to assess plan deliverability and clinical applicability of VMAT-HF as compared to VMAT-FF.ResultsWhile VMAT-HF used 60%–70% less intensity modulation complexity than VMAT-FF, it showed superior dose conformity. The small intestine and colon in VMAT-HF showed a noticeable reduction in the irradiated volumes of up to 35% and 15%, respectively, at an intermediate dose of 20–45 Gy. The small intestine showed statistically significant dose sparing at the volumes that received a dose from 15 to 45 Gy. Such a dose reduction for the small intestine and colon in VMAT-HF presented a significant NTCP reduction from that in VMAT-FF. Without sacrificing the beam delivery efficiency, VMAT-HF achieved effective OAR dose reduction in dose–volume histograms.ConclusionsVMAT-HF led to deliver conformal doses with effective gastrointestinal-OAR dose sparing despite using less modulation complexity. The dose of VMAT-HF was delivered with the same beam-on time with VMAT-FF but precise MLC leaf motions. The VMAT-HF potentially can play a valuable role in reducing OAR toxicities associated with WPRT.

Combining convex hull and directed graph for fast and accurate ellipse detection

Detecting ellipses from images is a fundamental task in many computer vision applications. However, due to the complexity of real-world scenarios, it is still a challenge to detect ellipses accurately and efficiently. In this paper, we propose a novel method to tackle this problem based on the fast computation of convex hull and directed graph, which achieves promising results on both accuracy and efficiency. We use Depth-First-Search to extract branch-free curves after adaptive edge detection. Line segments are used to represent the curvature characteristic of the curves, followed by splitting at sharp corners and inflection points to attain smooth arcs. Then the convex hull is constructed, together with the distance, length, and direction constraints, to find co-elliptic arc pairs. Arcs and their connectivity are encoded into a sparse directed graph, and then ellipses are generated via a fast access of the adjacency list. Finally, salient ellipses are selected subject to strict verification and weighted clustering. Extensive experiments are conducted on eight real-world datasets (six publicly available and two built by ourselves), as well as five synthetic datasets. Our method achieves the overall highest F-measure with competitive speed compared to representative state-of-the-art methods.

Rainbow Triangles in Arc-Colored Tournaments

Let \(T_{n}\) be an arc-colored tournament of order n. The maximum monochromatic indegree \(\varDelta ^{-mon}(T_{n})\) (resp. outdegree \(\varDelta ^{+mon}(T_{n})\)) of \(T_{n}\) is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of \(T_{n}\). The irregularity \(i(T_{n})\) of \(T_{n}\) is the maximum difference between the indegree and outdegree of a vertex of \(T_{n}\). A subdigraph H of an arc-colored digraph D is called rainbow if each pair of arcs in H have distinct colors. In this paper, we show that each vertex v in an arc-colored tournament \(T_{n}\) with \(\varDelta ^{-mon}(T_n)\le \varDelta ^{+mon}(T_n)\) is contained in at least \(\frac{\delta (v)(n-\delta (v)-i(T_n))}{2}-[\varDelta ^{-mon}(T_{n})(n-1)+\varDelta ^{+mon}(T_{n})d^+(v)]\) rainbow triangles, where \(\delta (v)=\min \{d^+(v), d^-(v)\}\). We also give some maximum monochromatic degree conditions for \(T_{n}\) to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible.

Double-sided arc welding with multiple electrodes of vertical joints of steel tanks

The increase of the welding productivity is still of immediate interest, which in turn requires the development and implementation of more high-performance welding methods that ensure a consistently high quality of welded joints. One of the ways to further increase the productivity of arc welding processes in the construction of tanks is the industrial use of automatic double-sided arc welding with multiple electrodes in separate weld pools. To establish the relationship between the parameters of the welding mode, energy characteristics, as well as the distance between the torches and their angle of inclination relative to the groove, a physical and mathematical model of the production of the welding joint when four arcs participate in pairs from both sides of the joint has been developed. Thus, computer modeling showed that during weld formation with the movement of the electrode down (vertical down), the possibility of supplying each pair of arcs with pulsed current should be taken into account. During groove filling with the movement of the electrodes up (vertical up), their lateral oscillations and delays at the edges should be carried out according to an aperiodic law in order to minimize the effects of ‘magnetic blow’.

COMPUTER ENGINEERING ANALYSIS OF DOUBLE-SIDED MULTI-ARC WELDING OF VERTICAL JOINTS OF TANKS FOR STORING OIL AND OIL PRODUCTS

The possibilities of computer engineering analysis of special features of double-sided multi-arc welding of vertical joints of tanks with thick shells for storing oil and oil products are shown. The analysis was carried out on the basis of a numerical implementation of the unconventional physical and mathematical model of the formation of a melt pool and a weld seam with two pairs of arcs on each side of the joint with a double-sided bevel. It was established that during weld formation with the movement of the electrode down (vertical down), the possibility of supplying the first pair of arcs with pulsed current should be taken into account, and for the second pairs, the possibility of lateral oscillation of the arcs in the groove with delays at the edges. During groove filling with the movement of the electrodes up (vertical up), their lateral oscillations and delays at the edges should be carried out according to an aperiodic law in order to minimize the effects of “magnetic blow”. For root formation, it is recommended to use pulsed welding, and during groove filling - high current welding with low welding speed. The results of the study can be used in the formation of scientifically-based requirements for the welding process and equipment for double-sided multi-arc welding of the vertical joints of tanks for storing oil and oil products.

An observational study of post-operative volumetric modulated arc radiotherapy (VMAT) of breast cancer by immobilizing patients on All-in-One (AIO™) solution breast and lung board without using thermoplastic mask: risk of collision of gantry with couch exists if acquisition of Cone Beam Computed Tomography (CBCT) is attempted for set-up verification

Automatic cone beam computed tomography (CBCT)–based image matching for set-up verification is recommended as compared with 2D image match for post-operative local/loco-regional radiotherapy of breast cancer patients by volumetric modulated arc therapy (VMAT) technique. However, in a supine position, off-midline peripheral body clinical target volume (CTV) of unilateral breast cancer patients immobilized on breast and lung board of All-in-One (AIO) positioning system may necessitate augmented movement of the couch in ‘x’ and ‘z’ axes thereby raising the risk of collision of x-ray sources/detector system with couch. VMAT was planned by a pair of partial arcs for the whole target volume for seven consecutive post-operative breast cancer patients (five post-mastectomy and two post-breast conservation patients). Tattoo-based set-up by the shift of treatment table in x-, y-, and z-axis as determined by treatment planning system followed by x-rays with planar image acquisition and online 2D image matching with Digitally reconstructed radiographs (DRRs) was performed for set-up verification. In-room 360° rotation of x-ray source and detector system of a linear accelerator (linac) was performed before x-ray planar image acquisition. Completion of 360° in-room rotation of x-ray source and detector system of linac around the machine isocentre was not possible in six out of seven patients due to the possibility of collision of the gantry with contralateral side of the couch. Performing CBCT for generating 3D images for computed tomography (CT) reconstruction may not be practical for patient set-up verification of post-operative radiotherapy of unilateral breast cancer patients positioned supine on breast and lung board.

Osculating conic biarcs

A circular biarc can be defined by using two points K and L with their (oriented) tangents gK and gL as input. It is well-known that one can determine a one parametric set of circular arc pairs k,ℓ such that k starts at K with tangent gK, ℓ ends at L with tangent gL and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace the circle biarcs by pairs of conic arcs. It turns out that in this case we can prescribe a conic k0 with a point K on it, another conic ℓ0 with a point L on it, and moreover an intermediate point P to obtain a unique pair k,ℓ of conics such that k osculates k0 in K, ℓ osculates ℓ0 in L and k and ℓ osculate each other in P. This also confirms a result of H. Pottmann from 1991. We use our method to solve an interpolation task of Hermite type whose input consists of a series of points with their curvature circles and another series of intermediate points. The output is a GC2 spline curve with conic arc segments.

Sources of Bipolar Outflows and Model Challenges for the Egg Nebula

Abstract The striking features of the Egg Nebula—searchlight beams that illuminate a pair of bipolar lobes, quadrupolar outflows, and arcs imprinted on an expanding spherical envelope—have long defied a coherent explanation. Here we present images at high angular resolutions of emission from dust and dense molecular gas in the nebula. Away from the systemic velocity, the molecular gas primarily traces the walls of the bipolar lobes, where bipolar outflows collide with the envelope ejected by the progenitor AGB star. At and close to the systemic velocity, both the molecular gas and dust trace the walls of a cylindrical channel swept aside by these bipolar outflows; in addition, the dust also traces a channel orthogonal to the bipolar lobes, swept aside by the orthogonal set of bipolar outflows. Close to the center, the dense gas traces an expanding spherical shell, corresponding to a very dense final puff from the progenitor AGB star. The center of this shell lies within the channel aligned with the optical lobes but is displaced to the south of the channel orthogonal to these lobes, and it closely coincides with the illumination center of the nebula (location of post-AGB star), as well as the expansion center of the bipolar lobes. We highlight difficulties faced by models whereby a binary companion both induces arcs in the wind of the progenitor AGB star and gives rise to the quadrupolar outflows. Instead, we suggest that the addition of a second companion, thus forming a triple-star system, can naturally produce all the features seen.

Exact solution algorithms for the maximum flow problem with additional conflict constraints

We consider a variant of the maximum flow problem on a given directed graph where some arc pairs are incompatible or conflicting; in other words, they are not allowed to carry positive flow simultaneously. This problem, called the maximum flow problem with conflicts, is known to be strongly NP-hard. In this paper, we present mixed-integer linear programming formulations for the problem and develop exact solution methods based on Benders decomposition, branch-and-bound, and Russian Doll Search over the conflict graph which represents the conflict relations. The effectiveness of the proposed algorithms is tested on a large number of randomly generated instances. The results reveal that their performances are superior to solving the mixed-integer linear programming formulations with a commercial software.

New Constructions for the n-Queens Problem

Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function $$l:V(G)\longrightarrow \{1,2,\ldots ,|V(G)|\}$$ such that, for every pair of arcs in E(D), namely (u, v) and $$(u',v')$$ we have (i) $$l(u)+l(v)\ne l(u')+l(v')$$ and (ii) $$l(v)-l(u)\ne l(v')-l(u')$$. Similarly, if the two conditions are satisfied modulo $$n=|V(G)|$$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of 1-regular digraphs and the solutions of the (modular) n-queens problem. The $$\otimes _h$$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $$\otimes _h$$-product and some particular families of graphs. In this paper, we study some families of 1-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) n-queens problem in terms of the $$\otimes _h$$-product, which in some sense complements a previous result due to Polya.

Probing the Magnetic Structure of a Pair of Transpolar Arcs With a Solar Wind Pressure Step

AbstractWe present observations of the Northern Hemisphere auroras taken with the far ultraviolet cameras onboard the Imager for Magnetopause‐to‐Aurora Global Exploration spacecraft during a compression of the magnetosphere by a solar wind pressure step on 30 December 2001. The compression occurs during a period of northward interplanetary magnetic field which has given rise to the presence of a pair of transpolar arcs (TPAs) near the dawnside oval. The compression causes a brightening of the oval, from dayside to nightside over the course of 10 min, followed by a brightening of the midnight sector oval and TPAs from nightside to dayside, again over 10 min. We suggest that the brightening is caused by pitch angle scattering of particles trapped on closed magnetic field lines and that the sequence of the brightening tracks the solar wind pressure step as it progresses along the length of the magnetotail. Traveling at 600 km s , the step reaches up to 90 down‐tail over the period of brightening, suggesting that the magnetic field lines which map to the TPAs are closed and stretch almost this length down‐tail.

Scattering of surface waves by a pair of asymmetric thin elliptic arc shaped plates with variable permeability

Abstract We consider the interaction between incident surface water waves and two asymmetric curved porous plates which assume the shape of ellipse. The permeability of the plates is not uniform. This is assumed to vary along the circumference of the elliptical arc. Two coupled integral equations are derived for the problem and are solved by employing an appropriate algorithm. Using these solutions, the numerical estimates for the reflection and the transmission coefficients, amplitudes of the hydrodynamic forces and energy dissipation coefficient are computed. The accuracy of the numerical results is established through an energy identity valid for permeable plates, and by comparing them with the published ones. A selection of new results is also presented for illustrative purposes. Importantly, in the present paper a mathematical method is developed to solve the problem of wave interaction with non-vertical (elliptic) plates with variable permeability. The derived numerical results indicate that the system of dual elliptic plates serves as a suitable model for a wave lens.