Unique Vertex Research Articles

Quantum walks on blow-up graphs

A blow-up of n copies of a graph G is the graph obtained by replacing every vertex of G by an independent set of size n, where the copies of two vertices in G are adjacent in the blow-up if and only if they are adjacent in G. In this work, we characterize strong cospectrality, periodicity, perfect state transfer (PST) and pretty good state transfer (PGST) in blow-up graphs. We prove that if a blow-up admits PST or PGST, then n = 2. In particular, if G has an invertible adjacency matrix, then each vertex in a blow of two copies of G pairs up with a unique vertex to exhibit strong cospectrality. Under mild conditions, we show that periodicity (resp., almost periodicity) of a vertex in G guarantees PST (resp. PGST) between the two copies of the vertex in the blow-up. This allows us to construct new families of graphs with PST from graphs that do not admit PST. We also characterize PST and PGST in the blow-ups of complete graphs, paths, cycles and cones. Finally, while trees in general do not admit PST, we provide infinite families of stars and subdivided stars whose blow-ups admit PST.

Probing Inert Triplet Model at a multi-TeV muon collider via vector boson fusion with forward muon tagging

This study investigates the potential of a multi-TeV Muon Collider (MuC) for probing the Inert Triplet Model (ITM), which introduces a triplet scalar field with hypercharge Y = 0 to the Standard Model. The ITM stands out as a compelling Beyond the Standard Model scenario, featuring a neutral triplet T0 and charged triplets T±. Notably, T0 is posited as a dark matter (DM) candidate, being odd under a Z2 symmetry. Rigorous evaluations against theoretical, collider, and DM experimental constraints corner the triplet scalar mass to a narrow TeV-scale region, within which three benchmark points are identified, with T± masses of 1.21 TeV, 1.68 TeV, and 3.86 TeV, for the collider study. The ITM’s unique TTVV four-point vertex, differing from fermionic DM models, facilitates efficient pair production through Vector Boson Fusion (VBF). This characteristic positions the MuC as an ideal platform for exploring the ITM, particularly due to the enhanced VBF cross-sections at high collision energies. To address the challenge of the soft decay products of T± resulting from the narrow mass gap between T± and T0, we propose using Disappearing Charged Tracks (DCTs) from T± and Forward muons as key signatures. We provide event counts for these signatures at MuC energies of 6 TeV and 10 TeV, with respective luminosities of 4 ab−1 and 10 ab−1. Despite the challenge of beam-induced backgrounds contaminating the signal, we demonstrate that our proposed final states enable the MuC to achieve a 5σ discovery for the identified benchmark points, particularly highlighting the effectiveness of the final state with one DCT and one Forward muon.

On homology groups for pairwise comparisons method

In this study, we introduce pairwise comparisons matrix classification based on homology groups of graphs with unique vertices. Algebraic topology transforms a sequence of topological objects (such as graphs associated with pairwise comparison matrices) into algebraic objects such as homology groups. It is the first attempt to use this tool to classify matrices of pairwise comparisons based on the triads in which the inconsistency occurs. The Koczkodaj inconsistency indicator was used in this study.

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

Variational sparse diffusion and its application in mesh processing

PurposeThe primary objective of this study is to tackle the enduring challenge of preserving feature integrity during the manipulation of geometric data in computer graphics. Our work aims to introduce and validate a variational sparse diffusion model that enhances the capability to maintain the definition of sharp features within meshes throughout complex processing tasks such as segmentation and repair.Design/methodology/approachWe developed a variational sparse diffusion model that integrates a high-order L1 regularization framework with Dirichlet boundary constraints, specifically designed to preserve edge definition. This model employs an innovative vertex updating strategy that optimizes the quality of mesh repairs. We leverage the augmented Lagrangian method to address the computational challenges inherent in this approach, enabling effective management of the trade-off between diffusion strength and feature preservation. Our methodology involves a detailed analysis of segmentation and repair processes, focusing on maintaining the acuity of features on triangulated surfaces.FindingsOur findings indicate that the proposed variational sparse diffusion model significantly outperforms traditional smooth diffusion methods in preserving sharp features during mesh processing. The model ensures the delineation of clear boundaries in mesh segmentation and achieves high-fidelity restoration of deteriorated meshes in repair tasks. The innovative vertex updating strategy within the model contributes to enhanced mesh quality post-repair. Empirical evaluations demonstrate that our approach maintains the integrity of original, sharp features more effectively, especially in complex geometries with intricate detail.Originality/valueThe originality of this research lies in the novel application of a high-order L1 regularization framework to the field of mesh processing, a method not conventionally applied in this context. The value of our work is in providing a robust solution to the problem of feature degradation during the mesh manipulation process. Our model’s unique vertex updating strategy and the use of the augmented Lagrangian method for optimization are distinctive contributions that enhance the state-of-the-art in geometry processing. The empirical success of our model in preserving features during mesh segmentation and repair presents an advancement in computer graphics, offering practical benefits to both academic research and industry applications.

Pipelines and Beyond: Graph Types for ADTs with Futures

Parallel programs are frequently modeled as dependency or cost graphs, which can be used to detect various bugs, or simply to visualize the parallel structure of the code. However, such graphs reflect just one particular execution and are typically constructed in a post-hoc manner. Graph types , which were introduced recently to mitigate this problem, can be assigned statically to a program by a type system and compactly represent the family of all graphs that could result from the program. Unfortunately, prior work is restricted in its treatment of futures , an increasingly common and especially dynamic form of parallelism. In short, each instance of a future must be statically paired with a vertex name. Previously, this led to the restriction that futures could not be placed in collections or be used to construct data structures. Doing so is not a niche exercise: such structures form the basis of numerous algorithms that use forms of pipelining to achieve performance not attainable without futures. All but the most limited of these examples are out of reach of prior graph type systems. In this paper, we propose a graph type system that allows for almost arbitrary combinations of futures and recursive data types. We do so by indexing datatypes with a type-level vertex structure , a codata structure that supplies unique vertex names to the futures in a data structure. We prove the soundness of the system in a parallel core calculus annotated with vertex structures and associated operations. Although the calculus is annotated, this is merely for convenience in defining the type system. We prove that it is possible to annotate arbitrary recursive types with vertex structures, and show using a prototype inference engine that these annotations can be inferred from OCaml-like source code for several complex parallel algorithms.

Spectral Turán problems for intersecting even cycles

Let C2k1,2k2,…,2kt denote the graph obtained by intersecting t distinct even cycles C2k1,C2k2,…,C2kt at a unique vertex. In this paper, we determine the unique graph with maximum adjacency spectral radius among all graphs on n vertices that do not contain any C2k1,2k2,…,2kt as a subgraph, for n sufficiently large. When one of the constituent even cycles is a C4, our results improve upper bounds on the Turán numbers for intersecting even cycles that follow from more general results of Füredi [21] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Turán problems on forbidden even cycles C2k,k≥2 (see [8,36,46,47]).

Asymmetric Structure of Podophage GP4 Reveals a Novel Architecture of Three Types of Tail Fibers

Bacteriophage tail fibers (or called tail spikes) play a critical role in the early stage of infection by binding to the bacterial surface. Podophages with known structures usually possess one or two types of fibers. Here, we resolved an asymmetric structure of the podophage GP4 to near-atomic resolution by cryo-EM. Our structure revealed a symmetry-mismatch relationship between the components of the GP4 tail with previously unseen topologies. In detail, two dodecameric adaptors (adaptors I and II), a hexameric nozzle, and a tail needle form a conserved tail body connected to a dodecameric portal occupying a unique vertex of the icosahedral head. However, five chain-like extended fibers (fiber I) and five tulip-like short fibers (fiber II) are anchored to a 15-fold symmetric fiber-tail adaptor, encircling the adaptor I, and six bamboo-like trimeric fibers (fiber III) are connected to the nozzle. Five fibers I, each composed of five dimers of the protein gp80 linked by an elongated rope protein, are attached to the five edges of the tail vertex of the icosahedral head. In this study, we identified a new structure of the podophage with three types of tail fibers, and such phages with different types of fibers may have a broad host range and/or infect host cells with considerably high efficiency, providing evolutionary advantages in harsh environments.

Unique eccentric point graphs and their eccentric digraphs

We study graph-theoretic properties of eccentric digraphs of unique eccentric point graphs (shortly, uep-graphs). The latter are the connected graphs in which every vertex has a unique eccentric vertex. In particular, we characterize uep-graphs and the corresponding eccentric digraphs in the following classes: self-centered graphs having the number of vertices twice as diameter, block graphs, and graphs with diameter three. Also, we obtain non-trivial properties of weak components in eccentric digraphs of uep-graphs with diameter four and pose several open questions in this direction.

Structure and proposed DNA delivery mechanism of a marine roseophage

Tailed bacteriophages (order, Caudovirales) account for the majority of all phages. However, the long flexible tail of siphophages hinders comprehensive investigation of the mechanism of viral gene delivery. Here, we report the atomic capsid and in-situ structures of the tail machine of the marine siphophage, vB_DshS-R4C (R4C), which infects Roseobacter. The R4C virion, comprising 12 distinct structural protein components, has a unique five-fold vertex of the icosahedral capsid that allows genome delivery. The specific position and interaction pattern of the tail tube proteins determine the atypical long rigid tail of R4C, and further provide negative charge distribution within the tail tube. A ratchet mechanism assists in DNA transmission, which is initiated by an absorption device that structurally resembles the phage-like particle, RcGTA. Overall, these results provide in-depth knowledge into the intact structure and underlining DNA delivery mechanism for the ecologically important siphophages.

Enumeration of Subperiodic Zeolitic Tetrahedral Frameworks. III. Layers, Rods, Tubes, and Clusters

We describe a new database of 4-coordinated hypothetical “subperiodic” zeolitic frameworks, comprising 0-periodic (clusters), 1-periodic (rods and tubes), and 2-periodic (layers) structures. The structures are found by a combinatorial search over 4-coordinated directed graphs, given the space group and the number of unique tetrahedral vertices, as in our earlier database. However, the filters are now reversed to reject the 3-periodic zeolites. Being fully 4-coordinated, these subperiodic structures require no hydroxyl terminations at their free surfaces when realized as SiO2 silicates. The layers closely resemble thin zeolites and are particularly interesting candidates for synthesis. The layer structures offer a rich library of structural possibilities for low-energy termination of the free surfaces of the known zeolites while maintaining full tetrahedral coordination. We present two examples of possible (001)-surface reconstructions for the LTL framework.

Planar median graphs and cubesquare-graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph G is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in G, and also in terms of subgraphs of G that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of G. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square-graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of “square-boundary” amalgamations. These considerations also lead to an O(nlogn)-time recognition algorithm to compute a decomposition of a planar median graph with n vertices into cubes and square-graphs.

Prime-valent symmetric graphs with a quasi-semiregular automorphism

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]).Let Γ be a connected symmetric graph of prime valency p≥5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph Cay(M,S) such that M is a 2-group admitting a fixed-point-free automorphism of order p with S as an orbit of involutions, or Γ is a normal N-cover of a T-arc-transitive graph of valency p admitting a quasi-semiregular automorphism, where T is a non-abelian simple group and N is a nilpotent group. Further, for p=5 a complete classification of graphs Γ such that either Aut(Γ) has a solvable arc-transitive subgroup or Γ is T-arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given.

RADT-31. 68GA-DOTATATE PET VERSUS MRI-BASED TREATMENT PLANNING FOR POST-OPERATIVE AND INTACT MENINGIOMA: A DOSIMETRIC ANALYSIS

Abstract Background Nearly all meningiomas express somatostatin receptor 1/2 (SSTR1/SSTR2); therefore, SSTR ligands such as 68Ga-DOTATATE can be utilized for meningioma radiotherapy treatment planning. Incorporation of 68Ga-DOTATATE PET assists with target delineation and may reduce the dose to organs-at-risk (OARs). We hypothesize that 68Ga-DOTATATE PET-based treatment plans will reduce dose to OARs when compared to MRI-based plans. METHODS All treatment plans were rendered on computed tomography (CT) planning datasets, using RapidArc and 6MV photon energy. Two arcs were positioned on the coplanar axis with alternated collimator settings of 0 and 90 degrees to promote MLC sparing with a third unique vertex arc chosen independently to spare normal brain tissue and associated organs at risk (OAR). For MRI structures, a 1 cm expansion from the gross tumor volume (GTV) created a clinical treatment volume (CTV)60. A 2 cm expansion created a CTV54. For PET structures, a 1 cm expansion from the GTV created a CTV60. A 3 mm isotropic expansion created planning treatment volumes (PTVs). Plans were optimized such that Dmax to brainstem remained limited to 60Gy or less and Dmax to optic structures was limited to 54Gy or less. RESULTS 18 meningioma patients were included (9 post-operative, 9 intact). The mean PTV volume using MRI was 258 ccs (306 ccs for post-operative, 210 ccs for intact), and the mean PTV volume using PET was 60 ccs (97 ccs for post-operative, 91 ccs for intact). The mean radiation dose to both optic nerves, the optic chiasm, both hippocampi, and the brainstem was reduced favoring PET-based treatment plans for both post-operative and intact patients. CONCLUSION Our study shows that incorporation of 68Ga-DOTATATE PET can reduce dose to OARs for post-operative and intact patients, and this difference may translate to reduced toxicity. 68Ga-DOTATATE PET guided radiation treatments are worthy of future prospective investigation.

68Ga-DOTATATE PET vs. MRI-Based Treatment Planning for Post-Operative and Intact Meningioma: A Dosimetric Analysis

<h3>Purpose/Objective(s)</h3> The current standard for radiation treatment planning involves MRI-based image guidance to define intact disease and residual tumor. Because nearly all meningiomas express somatostatin receptor 1/2 (SSTR1/SSTR2), SSTR ligands such as ⁶⁸Ga-DOTATATE are being explored for meningioma radiotherapy treatment planning. Incorporation of ⁶⁸Ga-DOTATATE PET can assist with radiation target volume delineation and reduce the dose to organs-at-risk (OARs). We hypothesize that ⁶⁸Ga-DOTATATE PET-based treatment plans will reduce dose to OARs when compared to MRI-based plans. <h3>Materials/Methods</h3> For each patient, 7 blinded physician contours were retrospectively performed on MRI and PET. A representative MRI and PET physician contour was chosen. All treatment plans were rendered on computed tomography (CT) planning datasets, using VMAT and 6MV photon energy. Each arrangement consisted of 3 independent partial arcs. Two arcs were positioned on the coplanar axis with alternated collimator settings of 0 and 90 degrees to promote MLC sparing with a third unique vertex arc chosen independently to spare normal brain tissue and associated organs at risk (OAR). For MRI structures, a 1 cm expansion from the gross tumor volume (GTV) was used to create a clinical treatment volume (CTV) 60. A 2 cm expansion was used to create a CTV 54. For PET structures, a 1 cm expansion from the GTV was used to create a CTV 60. All CTVs were cropped from outside the brain and a 3 mm isotropic expansion was used to create planning treatment volumes (PTVs). Each plan was prescribed to 95% coverage of prescription dose to the high risk PTV, and dosimetric maximum points to 0.03cc (Dmax) were evaluated for brainstem, optic chiasm, and both optic nerves. Plans were optimized such that Dmax to brainstem remained limited to 60 Gy or less and Dmax to optic structures was limited to 54 Gy or less. <h3>Results</h3> 18 meningioma patients were included (9 post-operative, 9 intact). The mean PTV volume on MRI was 258 ccs (306 ccs for post-operative, 210 ccs for intact), and the mean PTV volume on PET was 60 ccs (97 ccs for post-operative, 91 ccs for intact). The mean radiation dose was reduced to critical structures (Table 1). The mean brain dose was reduced by an average of 42% in PET-based plans. <h3>Conclusion</h3> ⁶⁸Ga-DOTATATE PET has potential to reduce treatment volume and dose to OARs for patients diagnosed with a meningioma. Our study shows that incorporation of ⁶⁸Ga-DOTATATE PET is especially useful in both the post-operative and intact setting and may reduce treatment toxicity. PET guided radiation treatments are worthy of future prospective investigation.

Spatial search via an interpolated memoryless walk

The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked vertex on a two-dimensional lattice. Our walk is based on the construction proposed by Falk, which tessellates the lattice with squares of size $2\ifmmode\times\else\texttimes\fi{}2$. Our walk uses minimal memory, $O(\sqrt{NlogN})$ applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a self-loop to the marked vertex---a technique we adapt from interpolated walks. We prove that with our explicit choice of self-loop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one.

Asymmetric RNA Egress Site in Expanded RNA Virus by Cryo‐EM and HDXMS

RNA viruses are metastable macromolecular assemblies that sense their environment through dynamic breathing motions. While surfaces of viral particles have been well characterized by cryo‐EM, dynamic genome‐capsid cores are poorly understood. To address this gap in understanding, we have applied hydrogen deuterium exchange mass spectrometry (HDXMS) together with cryo‐EM to Turnip Crinkle Virus (TCV) as a model system to study RNA genome‐capsid interactions and their importance in environmental sensing and programmed viral disassembly in favorable host environments. TCV undergoes expansion upon abstraction of Ca2+ ions, associated with entry into a plant host cell. We have mapped changes upon expansion in the viral capsid. HDXMS analysis reveals two conformations of the coat protein R‐domain regions in the native state: a minor (~5%) RNA‐bound and a major (~95%) unbound R‐domain. A larger fraction of the R‐domain is bound (~15%) more tightly to RNA in the expanded state, indicating expansion is accompanied by changes in conformation of the RNA‐coat protein core. Cryo‐EM of expanded TCV reveals that the newly formed strong interactions between the capsid proteins and the RNA genome correspond to the asymmetric ribonucleoprotein core packing observed near a unique 5‐fold vertex in the expanded TCV particle. This 5‐fold vertex is the specific point of egress for the genomic RNA into the host. Far from being passively package contents, our results highlight an active role of the genomic RNA‐protein core as an environmental sensor which controls capsid uncoating and RNA release.

Constrained shortest path tour problem: Branch-and-Price algorithm

The constrained shortest path tour problem consists, given a directed graph G=(V∪{s,t},A), an ordered set of disjoint vertex subsets T={T1,…,Tk} and a length function c:A→R+, in finding a path between s and t of minimum length in G intersecting every subset of T in the given order such that each arc is visited at most once. In this paper, we first show that this problem is NP-Hard even in the most particular case when T contains only one subset with a unique vertex. Then, we introduce a new mathematical model for the problem that helps its decomposition and develop an efficient Branch-and-Price algorithm. We demonstrate that it can easily be applied to several problem variants. Finally, we present extensive computational results with a benchmark against the state of the art Branch-and-Bound algorithm, called B&Bnew, from Ferone et al. (2020). On a diverse set of instances, we show that our algorithm significantly decreases the worst computational time while the ranking of algorithms for the average varies over instances.

Cryo-EM Structure of a Kinetically Trapped Dodecameric Portal Protein from the Pseudomonas-phage PaP3

Portal proteins are dodecameric assemblies that occupy a unique 5-fold vertex of the icosahedral capsid of tailed bacteriophages and herpesviruses. The portal vertex interrupts the icosahedral symmetry, and in vivo, its assembly and incorporation in procapsid are controlled by the scaffolding protein. Ectopically expressed portal oligomers are polymorphic in solution, and portal rings built by a different number of subunits have been documented in the literature. In this paper, we describe the cryo-EM structure of the portal protein from the Pseudomonas-phage PaP3, which we determined at 3.4 Å resolution. Structural analysis revealed a dodecamer with helical rather than rotational symmetry, which we hypothesize is kinetically trapped. The helical assembly was stabilized by local mispairing of portal subunits caused by the slippage of crown and barrel helices that move like a lever with respect to the portal body. Removing the C-terminal barrel promoted assembly of undecameric and dodecameric rings with quasi-rotational symmetry, suggesting that the barrel contributes to subunits mispairing. However, ΔC-portal rings were intrinsically asymmetric, with most particles having one open portal subunit interface. Together, these data expand the structural repertoire of viral portal proteins to Pseudomonas-phages and shed light on the unexpected plasticity of the portal protein quaternary structure.

Fast Neighborhood Rendezvous

In the rendezvous problem, two computing entities (called \emph{agents}) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous \emph{neighborhood rendezvous problem}, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in $O(\Delta)$ rounds ($\Delta$ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in $o(\Delta)$ rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of $O(\sqrt{n})$ rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $\tilde{O}(\sqrt{n\Delta/\delta} + n/\delta)$ rounds for graphs of the minimum degree larger than $\sqrt{n}$, where $n$ is the number of vertices in the graph, and $\delta$ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is $O(n)$, and achieves $\tilde{O}\left( \frac{n}{\sqrt{\delta}} \right)$-round time complexity without using whiteboards. These algorithms attain $o(\Delta)$-round complexity in the case of $\delta = {\omega}(\sqrt{n} \log n)$ and $\delta = \omega(n^{2/3} \log^{4/3} n)$ respectively.