Subset Of Edges Research Articles

On Interdicting Dense Clusters in a Network

Given a vertex-weighted undirected graph with blocking costs of its vertices and edges, we seek a minimum cost subset of vertices and edges to block such that the weight of any γ-quasi-clique in the interdicted graph is at most some predefined threshold parameter. The value of [Formula: see text] specifies the edge density of cohesive vertex groups of interest in the network. The considered weighted γ-quasi-clique interdiction problem can be viewed as a natural generalization of several variations of the clique blocker problem previously studied in the literature. From the application perspective, this setting is primarily motivated by the problem of disrupting adversarial (“dark”) networks (e.g., social or communication networks), where γ-quasi-cliques represent “tightly knit” groups of adversaries that we aim to dismantle. We first address the theoretical computational complexity of the problem. We then exploit some basic characterization of its feasible solutions to derive a linear integer programming (IP) formulation. This linear IP model can be solved using a lazy-fashioned branch-and-cut scheme. We also propose a combinatorial branch-and-bound algorithm for solving this problem. The computational performance of the developed exact solution schemes is studied using a test bed of randomly generated and real-life networks. Finally, some interesting insights and observations are also provided using a well-known example of a terrorist network. History: Accepted by Russel Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of S. Butenko was partially supported by the Air Force Office of Scientific Research under Award FA9550-23-1-0300. The work of O. A. Prokopyev was partially supported by the Office of Naval Research under Award ONR N00014-22-1-2678. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0027 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0027 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . The online appendix is available at https://doi.org/10.1287/ijoc.2023.0027 .

Realizations of Multiassociahedra via Rigidity

AbstractLet $$\Delta _{k}(n)$$ Δ k ( n ) denote the simplicial complex of $$(k+1)$$ ( k + 1 ) -crossing-free subsets of edges in $${\left( {\begin{array}{c}[n]\\ 2\end{array}}\right) }$$ [ n ] 2 . Here $$k,n\in \mathbb {N}$$ k , n ∈ N and $$n\ge 2k+1$$ n ≥ 2 k + 1 . Jonsson (2003) proved that [neglecting the short edges that cannot be part of any $$(k+1)$$ ( k + 1 ) -crossing], $$\Delta _{k}(n)$$ Δ k ( n ) is a shellable sphere of dimension $$k(n-2k-1)-1$$ k ( n - 2 k - 1 ) - 1 , and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are $$n\le 2k+3$$ n ≤ 2 k + 3 (Pilaud and Santos, Eur J Comb. 33(4):632–662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $$\Delta _{k}(n)$$ Δ k ( n ) as a polytope for $$(k,n)\in \{(2,9), (2,10) , (3,10)\}$$ ( k , n ) ∈ { ( 2 , 9 ) , ( 2 , 10 ) , ( 3 , 10 ) } . We also realize it as a simplicial fan for all $$n\le 13$$ n ≤ 13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for $$k\ge 3$$ k ≥ 3 and $$n\ge 2k+6$$ n ≥ 2 k + 6 no choice of points can realize $$\Delta _{k}(n)$$ Δ k ( n ) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

12531 Long-term Safety And Efficacy Of Once-daily Oral Paltusotine In The Treatment Of Patients With Acromegaly: Update From Acrobat Advance

Abstract Disclosure: M. Gadelha: Consulting Fee; Self; Crinetics, Ipsen, Novo Nordisk, and Recordati. Grant Recipient; Self; Crinetics and Recordati. Speaker; Self; Ipsen, Novo Nordisk, and Recordati. H.S. Randeva: None. M.B. Gordon: Consulting Fee; Self; Crinetics Pharmaceuticals, HRA Pharma, Novo Nordisk, and Recordati Rare Diseases. Grant Recipient; Self; Ascendis, Camurus, Chiasma, Corcept, Crinetics, Ipsen, Novartis, Novo Nordisk, Opko, Pfizer, and Strongbridge. M. Doknic: Research Investigator; Self; Crinetics Pharmaceuticals. Speaker; Self; Merck, Novartis, Novo Nordisk, Pfizer, and Sandoz. E. Mezosi: Research Investigator; Self; Crinetics Pharmaceuticals. M. Toth: Consulting Fee; Self; Pfizer, Ipsen, Novo Nordisk, and Recordati. Research Investigator; Self; Crinetics Pharmaceuticals. Other; Self; Pfizer, Ipsen, Novo Nordisk, and Recordati. C.L. Boguszewski: Advisory Board Member; Self; Novo Nordisk and Recordati. Consulting Fee; Self; Ipsen, Novo Nordisk, and Recordati. Other; Self; Ipsen, Novo Nordisk, and Recordati. C. Davidson: Employee; Self; Crinetics Pharmaceuticals. C.T. Ferrara-Cook: Employee; Self; Crinetics Pharmaceuticals. A. Casagrande: Employee; Self; Crinetics Pharmaceuticals. A. Krasner: Employee; Self; Crinetics Pharmaceuticals. Paltusotine is a non-peptide, highly selective SST2 receptor agonist in development as a once-daily, oral treatment for patients with acromegaly or carcinoid syndrome. ACROBAT Advance is an ongoing, 6-year, single-arm, open-label extension study of paltusotine in the treatment of patients with acromegaly. Interim results as the first enrolled patients approach 4 years of treatment are reported here. Enrolled patients had completed either the ACROBAT Edge or Evolve phase 2 parent studies. In Edge, at enrollment all patients were candidates for combination drug therapy: either sub-optimally controlled on an injected SRL (octreotide or lanreotide) alone or in combination with cabergoline, or required combination therapy or pasireotide to achieve normal IGF-I levels. In Evolve, enrolled patients had normal IGF-I levels on injected SRL monotherapy. When the Advance study was initiated, paltusotine was formulated as a capsule (dose range, 10-40 mg); all patients were switched to the tablet formulation (dose range, 20-60 mg) during the third year of the study. As of this analysis, all patients had at least 2 assessments after switching to tablet formulation. Adjunctive treatment with cabergoline or pegvisomant was allowed in patients who did not attain normal IGF-I levels on the maximum dose of paltusotine. Forty-three patients were enrolled in Advance (Edge, n=32; Evolve, n=11; 88% of eligible patients): at baseline, mean (±SD) age 53.0±11.6 years, 56% female, 86% previous pituitary surgery, and none had prior radiotherapy. IGF-I control in Edge and Evolve subsets remained stable at parent study baseline values. For all patients pooled, median (IQR) IGF-I levels were 1.15× ULN (0.84, 1.46; n=43) at parent study baseline; in Advance, 1.14× ULN (0.89, 1.29; n=40), 1.06× ULN (0.87, 1.24; n=35), and 1.08× ULN (0.87, 1.57; n=10) at months 12, 24, and 42, respectively. As expected, patients receiving adjunctive medication during Advance largely derived from the Edge study. Acromegaly symptoms, as measured using the patient-reported Acromegaly Symptom Diary (score range, 0-70; higher values indicate greater symptom burden), were stably controlled: median (IQR) score of 8.6 (3.6, 20.1; n=21) at parent study baseline; in Advance, 10.5 (5.0, 18.5; n=40), 10.0 (5.0, 25.0; n=34), and 13.5 (6.0, 22.0; n=10) at months 12, 24, and 42, respectively. The most common AEs reported through month 42 were arthralgia (37.2%), headache (30.2%), and fatigue (23.3%). One serious drug-related AE (cholelithiasis) was reported. Of the 8 patients who discontinued the study, 2 were due to AEs (mild or moderate). Glycemic control, as measured by HbA1c, was stable during paltusotine treatment. In conclusion, long-term results show that once-daily oral paltusotine treatment was well tolerated, with stable biochemical and symptom control relative to that observed with injected SRLs. Support: Crinetics Pharmaceuticals. Presentation: 6/3/2024

Distributed fusion-based algorithms for learning high-dimensional Bayesian Networks: Testing ring and star topologies

Learning Bayesian Networks (BNs) from high-dimensional data is a complex and time-consuming task. Although there are approaches based on horizontal (instances) or vertical (variables) partitioning in the literature, none can guarantee the same theoretical properties as the Greedy Equivalence Search (GES) algorithm, except those based on the GES algorithm itself. This paper proposes a parallel distributed framework that uses GES as its local learning algorithm, obtaining results similar to those of GES and guaranteeing its theoretical properties but requiring less execution time. The framework involves splitting the set of all possible edges into clusters and constraining each framework node to only work with the received subset of edges. The global learning process is an iterative algorithm that carries out rounds until a convergence criterion is met. We have designed a ring and a star topology to distribute node connections. Regardless of the topology, each node receives a BN as input; it then fuses it with its own BN model and uses the result as the starting point for a local learning process, limited to its own subset of edges. Once finished, the result is then sent to another node as input. Experiments were carried out on a large repertory of domains, including large BNs up to more than 1000 variables. Our results demonstrate our proposal's effectiveness compared to GES and its fast version (fGES), generating high-quality BNs in less execution time.

An Efficient Tour Construction Heuristic for Generating the Candidate Set of the Traveling Salesman Problem with Large Sizes

In this paper, we address the challenge of creating candidate sets for large-scale Traveling Salesman Problem (TSP) instances, where choosing a subset of edges is crucial for efficiency. Traditional methods for improving tours, such as local searches and heuristics, depend greatly on the quality of these candidate sets but often struggle in large-scale situations due to insufficient edge coverage or high time complexity. We present a new heuristic based on fuzzy clustering, designed to produce high-quality candidate sets with nearly linear time complexity. Thoroughly tested on benchmark instances, including VLSI and Euclidean types with up to 316,000 nodes, our method consistently outperforms traditional and current leading techniques for large TSPs. Our heuristic’s tours encompass nearly all edges of optimal or best-known solutions, and its candidate sets are significantly smaller than those produced with the POPMUSIC heuristic. This results in faster execution of subsequent improvement methods, such as Helsgaun’s Lin–Kernighan heuristic and evolutionary algorithms. This substantial enhancement in computation time and solution quality establishes our method as a promising approach for effectively solving large-scale TSP instances.

Hamiltonian cycles of balanced hypercube with disjoint faulty edges

The balanced hypercube BHn, a variant of the hypercube, is a novel interconnection network topology for massive parallel systems. It is showed in [Theor. Comput. Sci. 947 (2023) 113708] that for any edge subset F of BHn there exists a fault-free Hamiltonian cycle in BHn−F for n≥2 with |F|≤5n−7 if the degree of every vertex in BHn−F is at least two and there exist no f4-cycles in BHn−F. In this paper, we consider the existence of Hamiltonian cycles of BHn when F is a matching (a set of disjoint edges), and show that each edge e∉F lies on a fault-free Hamiltonian cycle of BHn−F with n≥2. The number of faulty edges in F can be up to 22n−1, which is exponential to the dimension n.

Network design with vulnerability constraints and probabilistic edge reliability

AbstractThe network design problem with vulnerability constraints and probabilistic edge reliability (NDPVC‐PER) is an extension of the NDPVC obtained by additionally considering edge reliability. We consider the design of a telecommunication network in which every origin‐destination pair is connected by a hop‐constrained primal path, and by a hop‐constrained backup path when certain edges in the network fail. The edge failures occur with respect to their reliability, and the network is designed by considering a minimum reliability level. Therefore, a hop‐constrained backup path must be built by considering all simultaneous edge failures that have a certain probability of realization. While there exist models to solve the NDPVC without enumerating all edge subsets, edge reliability cannot be dealt with by applying the techniques applied to the NDPVC. Therefore, we develop models based on a new concept of resilient length‐bounded cuts, and solve the NDPVC‐PER without edge set enumerations. We perform extensive testing of the model to determine the best performing settings, and demonstrate the computational efficiency of the developed model. Our findings on these instances show that, in the dataset considered in this study, increasing the reliability level from 90% to 95% increases the average cost only by 12.4%, while increasing it from 95% to 99% level yields a cost increase of 93.9%.

Locating Edge Domination Number of Some Classes of Claw-Free Cubic Graphs

Let G=V;E be a simple graph with vertex set V and edge set E. In a graph G, a subset of edges denoted by M is referred to as an edge-dominating set of G if every edge that is not in M is incident to at least one member of M. A set M⊆E is the locating edge-dominating set if for every two edges e1,e2∈E−M, the sets Ne1∩M and Ne2∩M are nonempty and different. The edge domination number γLG of G is the minimum cardinality of all edge-dominating sets of G. The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.

A Validation of the Phenomenon of Linearly Many Faults on Burnt Pancake Graphs with Its Applications

“Linearly many faults” is a phenomenon observed by Cheng and Lipták in which a specific structure emerges when a graph is disconnected and often occurs in various interconnection networks. This phenomenon means that if a certain number of vertices or edges are deleted from a graph, the remaining part either stays connected or breaks into one large component along with smaller components with just a few vertices. This phenomenon can be observed in many types of graphs and has important implications for network analysis and optimization. In this paper, we first validate the phenomenon of linearly many faults for surviving graph of a burnt pancake graph BPn when removing any edge subset with a size of approximately six times λ(BPn). For graph G, the ℓ-component edge connectivity denoted as λℓ(G) (resp., the ℓ-extra edge connectivity denoted as λ(ℓ)(G)) is the cardinality of a minimum edge subset S such that G−S is disconnected and has at least ℓ components (resp., each component of G−S has at least ℓ+1 vertices). Both λℓ(G) and eλ(ℓ)(G) are essential metrics for network reliability assessment. Specifically, from the property of “linearly many faults”, we may further prove that λ5(BPn)=λ(3)(BPn)+3=4n−3 for n⩾5; λ6(BPn)=λ(4)(BPn)+4=5n−4 and λ7(BPn)=λ(5)(BPn)+5=6n−5 for n⩾6.

Distance-Preserving Graph Compression Techniques

We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights and a subset of edges $E^{\prime} \subset E$, which needs to be removed from G (with their endpoints merged as a supernode). The goal is to redistribute the weights of the deleted edges in a way that minimizes the error. The error is defined as the sum of the absolute differences of the shortest path lengths between different pairs of nodes before and after contracting $E^{\prime}$. Based on this error function, we propose optimal approaches for merging any subset of edges in a path and a single edge in a tree. Previous works on graph compression techniques aimed at preserving different graph properties (such as the chromatic number) or solely focused on identifying the optimal set of edges to contract. However, our focus in this paper is on achieving optimal edge contraction (when the contracted edges are provided as input), specifically for weighted trees and paths.

Characterizing edge-based doubly resolving sets within circulant networks $ C_n(1, 2) $

<abstract><p>The focus of this article lies on the notion of the edge version of doubly resolving sets (EVDRSs) in circulant networks. EVDRSs refer to unique edge subsets that are necessary for identifying individual edges in a network and distinguishing them based on their edge distances to the elements of the EVDRS. The main objectives were to define the minimal size of EVDRSs for circulant networks $ C_n(1, 2) $ and to investigate their basic properties. The systematic research helped to achieve a new understanding of the existence, construction, and characterization of EVDRSs in circulant networks $ C_n(1, 2) $. It is established that the EVDRSs in the circulant network $ C_n(1, 2) $ are finite and are bounded by the order of the network. Among the numerous implications of these findings are those that refer to the design and optimization of distributed sensor networks, improving communication and network protocols, as well as tracking the spread of infectious diseases and epidemics over social networks. The application of the identified methodology helps improve the process of network optimization which contributes to the development of more effective and robust circulant-based structures.</p></abstract>

2-node-connectivity network design

We consider 2-connectivity network design problems in which we are given a graph and seek a min-size 2-connected subgraph that satisfies a prescribed property.•In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a minimum size edge subset of a specified edge set such that the augmented graph is 2-connected. We breach the natural approximation ratio 2 for this problem, and also for the more general Crossing Family Cover problem.•In the 2-Connected Dominating Set problem, we seek a minimum size 2-connected subgraph that dominates all nodes. We give the first non-trivial approximation algorithm with expected approximation ratio O(σ(n)⋅log3⁡n), where σ(n)=O(log⁡n⋅log⁡log⁡n⋅(log⁡log⁡log⁡n)3).The unifying technique of both results is a reduction to the Subset Steiner Connected Dominating Set problem. Such a reduction was known for edge-connectivity, and we extend it to 2-node connectivity problems. We show that the same method can be used to obtain easily polylogarithmic approximation ratios that are not too far from the best known ones for several other problems.

Characterizing signed mixed graphs with small eigenvalues

A mixed graph MG is obtained from a simple graph G by orienting an edge subset of G. A signed mixed graph is a mixed graph with arcs and edges signed + or −. The unit Eisenstein matrix (E-matrix for short) of a signed mixed graph was recently introduced by Wissing and van Dam [32]. This novel matrix is indexed by the vertices of the signed mixed graph, and the entry corresponding to a positive arc from u to v is equal to ω=1+i32 (and its symmetric entry is ω‾=1−i32); the entry corresponding to a negative arc is equal to −ω (and its symmetric entry is −ω‾); the entry corresponding to a positive edge is equal to 1; the entry corresponding to a negative edge is equal to −1; and 0 otherwise. In this paper, we study the spectral properties of this E-matrix. We characterize all the signed mixed graphs whose eigenvalues are contained in (−α,α) for α∈{2,3,2}.

Fine tuning a logical model of cancer cells to predict drug synergies: combining manual curation and automated parameterization

Treatment with combinations of drugs carries great promise for personalized therapy for a variety of diseases. We have previously shown that synergistic combinations of cancer signaling inhibitors can be identified based on a logical framework, by manual model definition. We now demonstrate how automated adjustments of model topology and logic equations both can greatly reduce the workload traditionally associated with logical model optimization. Our methodology allows the exploration of larger model ensembles that all obey a set of observations, while being less restrained for parts of the model where parameterization is not guided by biological data. We benchmark the synergy prediction performance of our logical models in a dataset of 153 targeted drug combinations. We show that well-performing manual models faithfully represent measured biomarker data and that their performance can be outmatched by automated parameterization using a genetic algorithm. Whereas the predictive performance of a curated model is strongly affected by simulated curation errors, data-guided deletion of a small subset of regulatory model edges can significantly improve prediction quality. With correct topology we find evidence of some tolerance to simulated errors in the biomarker calibration data, yet performance decreases with reduced data quality. Moreover, we show that predictive logical models are valuable for proposing mechanisms underpinning observed synergies. With our framework we predict the synergy of joint inhibition of PI3K and TAK1, and further substantiate this prediction with observations in cancer cell cultures and in xenograft experiments.

Theoretical and computational analysis of a new formulation for the Rural Postman Problem and the General Routing Problem

The Rural Postman Problem (RPP) is one of the most well-known problems in arc routing. Given an undirected graph, the RPP consists of finding a closed walk traversing and servicing a given subset of edges with minimum total cost. In the General Routing Problem (GRP), there is also a subset of vertices that must be visited. Both problems were introduced by Orloff and proved to be NP-hard. In this paper, we propose a new formulation for the RPP and the GRP using two sets of binary variables representing the first and second traversal, respectively, of each edge. We present several families of valid inequalities that induce facets of the polyhedron of solutions under mild conditions. Using this formulation and these families of inequalities, we propose a branch-and-cut algorithm, test it on a large set of benchmark instances, and compare its performance against the exact procedure that, as far as we know, produced the best results. The results obtained show that the proposed formulation is useful for solving undirected RPP and GRP instances of very large size.

Sharp thresholds in adaptive random graph processes

AbstractThe ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).

Demystifying Graph Sparsification Algorithms in Graph Properties Preservation

Graph sparsification is a technique that approximates a given graph by a sparse graph with a subset of vertices and/or edges. The goal of an effective sparsification algorithm is to maintain specific graph properties relevant to the downstream task while minimizing the graph's size. Graph algorithms often suffer from long execution time due to the irregularity and the large real-world graph size. Graph sparsification can be applied to greatly reduce the run time of graph algorithms by substituting the full graph with a much smaller sparsified graph, without significantly degrading the output quality. However, the interaction between numerous sparsifiers and graph properties is not widely explored, and the potential of graph sparsification is not fully understood. In this work, we cover 16 widely-used graph metrics, 12 representative graph sparsification algorithms, and 14 real-world input graphs spanning various categories, exhibiting diverse characteristics, sizes, and densities. We developed a framework to extensively assess the performance of these sparsification algorithms against graph metrics, and provide insights to the results. Our study shows that there is no one sparsifier that performs the best in preserving all graph properties, e.g. sparsifiers that preserve distance-related graph properties (eccentricity) struggle to perform well on Graph Neural Networks (GNN). This paper presents a comprehensive experimental study evaluating the performance of sparsification algorithms in preserving essential graph metrics. The insights inform future research in incorporating matching graph sparsification to graph algorithms to maximize benefits while minimizing quality degradation. Furthermore, we provide a framework to facilitate the future evaluation of evolving sparsification algorithms, graph metrics, and ever-growing graph data.

Link fault tolerance of BC networks and folded hypercubes on h-extra r-component edge-connectivity

Parallel and distributed systems play a significant role in high-performance computing, prompting us to investigate qualitative and quantitative metrics to indicate the fault tolerance and vulnerability of systems. Consider the setup where there are large-scale link malfunctions that disconnect the network and result in various components, each with multiple processors. In this paper, we propose and study the h-extra r-component edge-connectivity of a connected graph G, which is denoted by cλrh(G) and has not been addressed before. Let h and r be two positive integers with r≥2. An edge subset F⊆E(G) is said to be an h-extra r-component edge-cut of G, if any, G−F has at least r components and every component of G−F has at least h vertices. The cardinality of the minimum h-extra r-component edge-cut of G is the h-extra r-component edge-connectivity of G. Let h be a positive integer with the decomposition h=∑i=0t2ki, where ki>ki+1, 0≤i≤t−1. In this paper, we derive a lower bound for the exact value of h-extra 3-component edge-connectivity of BC networks Bn and demonstrate that it is tight for one member of Bn, hypercube Qn, in the interval 1≤h≤2⌊n2⌋−1−1, n≥4. This lower bound is also tight for the exact value of 2c-extra 3-component edge-connectivity of Bn with 0≤c≤n−2, n≥4. Specifically, cλ3h(Qn)=2nh−∑i=0tki2ki+1−∑i=0ti2ki+2−h for 1≤h≤2⌊n2⌋−1−1, n≥4 and cλ32c(Bn)=(2n−2c−1)2c for 0≤c≤n−2, n≥4. Exact value of h-extra 3-component edge-connectivity of n-dimensional folded hypercube FQn, cλ3h(FQn) is 2(n+1)h−∑i=0tki2ki+1−∑i=0ti2ki+2−h for 1≤h≤2⌈n2⌉−1−1, n≥4, and that of 2c-extra 3-component edge-connectivity of FQn, cλ32c(FQn), is (2n−2c+1)2c for 0≤c≤n−2, n≥4.

Extremal results on feedback arc sets in digraphs

AbstractFor an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.

The Super Restricted Edge-connectedness of Direct Product Graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. An edge subset [Formula: see text] is called a restricted edge-cut if [Formula: see text] is disconnected and has no isolated vertices. The restricted edge-connectivity [Formula: see text] of [Formula: see text] is the cardinality of a minimum restricted edge-cut of [Formula: see text] if it has any; otherwise [Formula: see text]. If [Formula: see text] is not a star and its order is at least four, then [Formula: see text], where [Formula: see text]. The graph [Formula: see text] is said to be maximally restricted edge-connected if [Formula: see text]; the graph [Formula: see text] is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from [Formula: see text]. The direct product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient condition for [Formula: see text] to be super restricted edge-connected, where [Formula: see text] is the complete graph on [Formula: see text] vertices.