Arbitrary Tree Research Articles

Remote State Estimation Under DoS Attacks in CPSs With Arbitrary Tree Topology: A Bayesian Stackelberg Game Approach

Remote State Estimation Under DoS Attacks in CPSs With Arbitrary Tree Topology: A Bayesian Stackelberg Game Approach

A novel design employing arbitrary tree structure for adaptive thrsholding discrete wavelet packet transformation

A novel design employing arbitrary tree structure for adaptive thrsholding discrete wavelet packet transformation

Synthesis of a quantum tree Weyl matrix

A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step.

Checking in polynomial time whether or not a regular tree language is deterministic top-down

It is well known that for a given bottom-up tree automaton it can be decided whether or not an equivalent deterministic top-down tree automaton exists. Recently it was claimed that such a decision can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their procedure and corresponding property is wrong. Here we address this mistake and present a correct property which allows to determine in polynomial time whether or not a given tree language can be recognized by a deterministic top-down tree automaton. Furthermore, our new property is stated for arbitrary deterministic bottom-up tree automata, and not only for minimal such automata (as before).

Three-vertex prime graphs and reality of trees

We continue the study of prime simple modules for quantum affine algebras from the perspective of q-fatorization graphs. In this paper we establish several properties related to simple modules whose q-factorization graphs are afforded by trees. The two most important of them are proved for type A. The first completes the classification of the prime simple modules with three q-factors by giving a precise criterion for the primality of a 3-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose q-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general.

Sharp bound on the truncated metric dimension of trees

A k-truncated resolving set of a graph is a subset S⊆V of its vertex set such that the vector (dk(s,v))s∈S is distinct for each vertex v∈V where dk(x,y)=min⁡{d(x,y),k+1} is the graph distance truncated at k+1. We think of elements of a k-truncated resolving set as sensors that can measure up to distance k. The k-truncated metric dimension (Tmdk) of a graph G is the minimum cardinality of a k-truncated resolving set of G. We give a sharp lower bound on Tmdk for any tree T in terms of its number of vertices |T| and the measuring radius k. Our result is that Tmdk(T)≥|T|⋅3/(k2+4k+3+1{k≡1(mod3)})+ck, disproving earlier conjectures by Frongillo et al. that suspected |T|/(⌊k2/4⌋+2k)+ck′ as general lower bound, where ck, ck′ are k-dependent constants. We provide a construction for trees with the largest number of vertices with a given Tmdk value. The proof that our optimal construction cannot be improved relies on edge-rewiring procedures of arbitrary (suboptimal) trees with arbitrary resolving sets, which reveal the structure of how small subsets of sensors measure and resolve certain areas in the tree that we call the attraction of those sensors. The notion of ‘attraction of sensors’ might be useful in other contexts beyond trees to solve related problems. We also provide an improved lower bound on Tmdk of arbitrary trees that takes into account the structural properties of the tree, in particular, the number and length of simple paths of degree-two vertices terminating in leaf vertices. This bound complements the result of the above-mentioned work of Frongillo et al., where only trees without degree-two vertices were considered, except the simple case of a single path.

Functional-Differential Operators on Geometrical Graphs with Global Delay and Inverse Spectral Problems

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a locally nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce globally nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area for further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.

A linear algorithm for the neighbor-component order connectivity of arbitrary trees

The graph parameter neighbor-connectivity was first investigated by Gunther and Hartnell in 1987 and provides important information on how reliable a network can be when failures of a node may impact its neighbors. In this model, the failure of a node causes the deletion of its closed neighborhood, i.e., the node and its adjacent neighbors as well. The minimum number of closed neighborhoods whose removal results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Since then component order connectivity models have emerged to address inadequacies in the traditional models of connectivity, namely, that in many real-world networks, when disconnecting a graph one may be left with components that are large enough to still be operable. By adapting neighbor-connectivity to a component order model, we introduced neighbor-component order connectivity, defined as the minimum number of closed neighborhoods that must be removed from a network to ensure all remaining components have order less than some given threshold value. Given a threshold value of one, the neighbor-component order connectivity of a graph is equivalent to the well-known parameter domination number of the graph. The problem of computing the domination number of an arbitrary graph is NP-hard, and computing neighbor-component order connectivity of a graph for an arbitrary threshold value is also NP-hard. Here, we present a linear-time algorithm for computing the neighbor-component order connectivity of an arbitrary tree for an arbitrary threshold value, thus generalizing the classic linear algorithm of Cockayne, Goodman, and Hedetniemi for the domination number of the tree.

Total Domination on Tree Operators

Let G be a graph with vertex set V and edge set E, a set Dsubseteq V is a total dominating set if every vertex vin V has at least one neighbor in D. The minimum cardinality among all total dominating sets is called the total domination number, and it is denoted by gamma _{t}(G). Given an arbitrary tree graph T, we consider some operators acting on this graph; texttt {S}(T),texttt {R}(T),texttt {Q}(T) and texttt {T}(T), and we give bounds of the total domination number of these new graphs using other parameters in the graph T. We also give the exact value of the total domination number in some of them.

Note on graph-based BCJ relation for Berends-Giele currents

Graph-based Bern-Carasso-Johansson (BCJ) relation for Berends-Giele currents in bi-adjoint scalar (BS) theory, which is characterized by connected tree graphs, was proposed in an earlier work. In this note, we provide a systematic study of this relation. We first prove the relations based on two special types of graphs: simple chains and star graphs. The general graph-based BCJ relation established by an arbitrary tree graph is further proved, through Berends-Giele recursion. When combined with proper off-shell extended numerators, this relation induces the graph-based BCJ relation for Berends-Giele currents in Yang-Mills theory. The corresponding relations for amplitudes are obtained via on-shell limits.

Software Defect Prediction and Analysis Using Enhanced Random Forest (extRF) Technique: A Business Process Management and Improvement Concept in IOT-Based Application Processing Environment

Software defect prediction is a thriving study area in the realm of software engineering and processing in the IOT-based environment. Defect prediction creates a list of defective source code artifacts so that quality assurance companies may successfully assign limited methods for certifying programming things by investing more effort into the bad source code. Defect prediction can assist estimate maintenance times, which can help with quality assurance, dependability, security, and cost reduction. Many predictions in IOT-based processing environment and business process management and enhancement challenges still exist in defect expectation ponders, and there are various noteworthy concerns. In addition, it is difficult to apply these methodologies practically because most of the investigations verified in open-source programming ventures with the goal that present forecast models might not work for other programming items including business programming. Investigating security issues in cross-project deformity expectation is required since if we have more accessible restrictive datasets, the assessment of forecast models will be more stable. In general, every defect is essential regarding quality, reliability, security, and cost-effectiveness. Therefore, an enhanced and improved maintenance schedule is required to acknowledge forecasting techniques. Therefore, in this article, we have evaluated different Semi-Supervised Learning (SSL) techniques, among which Extended Random Forest (extRF) technique is one for defective system prediction. The Extended Random Forest (extRF) technique is the extended form of the Random Forest (RF), which is a supervised learning technique into semi-supervised learning getting the hang of refining every arbitrary tree given an individual-training worldview. An enhancing technique is recommended, and a weighted mixture of irregular trees creates the final forecast results.

Efficient Isomorphism for $$S_d$$-Graphs and T-Graphs

An H-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró et al. (Discrete Mathematics 100:267–279, 1992). An H-graph is proper if the representing subgraphs of H can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for \(S_d\)-graphs and T-graphs, where \(S_d\) is the star with d rays and T is an arbitrary fixed tree. Answering an open problem of Chaplick et al. (2016, personal communication), we provide an FPT-time algorithm for testing isomorphism and computing the automorphism group of \(S_d\)-graphs when parameterized by d, which involves the classical group-computing machinery by Furst et al. (in Proceedings of 11th southeastern conference on combinatorics, graph theory, and computing, congressum numerantium 3, 1980). We also show that the isomorphism problem of \(S_d\)-graphs is at least as hard as the isomorphism problem of posets of bounded width, for which no efficient combinatorial-only algorithm is known to date. Then we extend our approach to an XP-time algorithm for isomorphism of T-graphs when parameterized by the size of T. Lastly, we contribute an FPT-time combinatorial algorithm for isomorphism testing in the special case of proper \(S_d\)- and T-graphs.

Flexibility Framework With Recovery Guarantees for Aggregated Energy Storage Devices

This paper proposes a framework for the procurement of flexibility reserve from aggregated storage fleets. It allows for arbitrary tree structures of aggregation hierarchy, as well as easily implementable disaggregation via broadcast dispatch. By coupling discharge and recovery modes, the proposed framework enables full-cycle capacity to be procured ahead of real time, with guaranteed recovery and exact accounting for losses. The set of feasible discharging requests is exactly encoded, so that there is no reduction in the ability to meet discharging signals, and recovery capabilities are parametrised as a single virtual battery. Included in this paper is a numerical demonstration of the construction of the constituent curves of the framework and the approach is also benchmarked against relevant alternatives.

The enumeration of extreme rigid honeycombs

Rigid honeycombs were introduced by Knutson et al. (J Am Math Soc 17:19–48, 2004), and they were shown in Bercovici et al. (J Funct Anal 258:1579–1627, 2010) to be sums of extreme rigid honeycombs, with uniquely determined summands up to permutations. Two extreme rigid honeycombs are essentially the same if they have proportional exit multiplicities and, up to this identification, there are countably many equivalence classes of such honeycombs. We describe two ways to approach the enumeration of these equivalence classes. The first method produces a (finite) list of all rigid tree honeycombs of fixed weight by looking at the locking patterns that can be obtained from a certain quadratic Diophantine equation. The second method constructs arbitrary rigid tree honeycombs from rigid overlays of two rigid tree honeycombs with strictly smaller weights. This allows, in principle, for an inductive construction of all rigid tree honeycombs starting with those of unit weight. We also show that some rigid overlays of two rigid tree honeycombs give rise to an infinite sequence of rigid tree honeycombs of increasing complexity but with a fixed number of nonzero exit multiplicities. This last result involves a new inflation/deflation construction that also produces other infinite sequences of rigid tree honeycombs.

Development of Algorithm for Calculating Data Packet Transmission Delay in Software-Defined Networks

The relevance of this type of network is associated with the development and improvement of protocols, methods, and tools to verify routing policies and algorithmic models describing various aspects of SDN, which determined the purpose of this study. The main purpose of this work is to develop specialized methods to estimate the maximum end-to-end delay during packet transmission using SDN infrastructure. The methods of network calculus theory are used to build a model for estimating the maximum transmission delay of a data packet. The basis for this theory is obtaining deterministic evaluations by analyzing the best and worst-case scenarios for individual parts of the network and then optimally combining the best ones. It was found that the developed method of theoretical evaluation demonstrates high accuracy. Consequently, it is shown that the developed algorithm can estimate SND performance. It is possible to conclude the configuration optimality of elements in the network by comparing the different possible configurations. Furthermore, the proposed algorithm for calculating the upper estimate for packet transmission delay can reduce network maintenance costs by detecting inconsistencies between network equipment settings and requirements. The scientific novelty of these results is that it became possible to calculate the achievable upper data delay in polynomial time even in the case of arbitrary tree topologies, but not only when the network handlers are located in tandem. Doi: 10.28991/ESJ-2022-06-05-010 Full Text: PDF

Optimal Broadcasting in Fully Connected Trees

Broadcasting is disseminating information in a network where a specific message must spread to all network vertices as quickly as possible. Finding the minimum broadcast time of a vertex in an arbitrary network is proven to be NP-complete. However, this problem is solvable for a few families of networks. In this paper, we present an optimal algorithm for finding the broadcast time of any vertex in a fully connected tree ([Formula: see text]) in [Formula: see text] time. An [Formula: see text] is formed by attaching arbitrary trees to vertices of a complete graph of size [Formula: see text] where [Formula: see text] is the total number of vertices in the graph.

The Edge General Position Problem

Given a graph G, the general position problem is to find a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is called a \(\mathrm{gp}\)-set of G, and its cardinality is the \(\mathrm{gp}\)-number, \(\mathrm{gp}(G)\), of G. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem.The edge general position number, \(\mathrm{gp_{e}}(G)\), is the size of a largest edge general position set of G. For r-dimensional hypercube \(Q_r\), it is proved that \(\mathrm{gp_{e}}(Q_r) = 2^r\), and for arbitrary tree T, it is shown that \(\mathrm{gp_{e}}(T)\) is the number of its leaves. The value of \(\mathrm{gp_{e}}(P_r\, \square \, P_s)\) is determined for every \(r,s\ge 2\). To derive these results, the theory of partial cubes is used. Mulder’s meta-conjecture on median graphs is also discussed along the way.

The Probability of Joint Monophyly of Samples of Gene Lineages for All Species in an Arbitrary Species Tree.

Monophyly is a feature of a set of genetic lineages in which every lineage in the set is more closely related to all other members of the set than it is to any lineage outside the set. Multiple sets of lineages that are separately monophyletic are said to be reciprocally monophyletic, or jointly monophyletic. The prevalence of reciprocal monophyly, or joint monophyly (JM), has been used to evaluate phylogenetic and phylogeographic hypotheses, as well as to delimit species. These applications often make use of a probability of JM under models of gene lineage evolution. Studies in coalescent theory have computed this JM probability for small numbers of separate groups in arbitrary species trees and for arbitrary numbers of separate groups in trivial species trees. In this study, generalizing existing results on monophyly probabilities under the multispecies coalescent, we derive the probability of JM for arbitrary numbers of separate groups in arbitrary species trees. We illustrate how our result collapses to previously examined cases. We also study the effect of tree height, sample size, and number of species on the probability of JM. We obtain relatively simple lower and upper bounds on the JM probability. Our results expand the scope of JM calculations beyond small numbers of species, subsuming past formulas that have been used in simpler cases.

Applications of Order Trees in Infinite Graphs

Traditionally, the trees studied in infinite graphs are trees of height at most ω, with each node adjacent to its parent and its children (and every branch of the tree inducing a path or a ray). However, there is also a method, systematically introduced by Brochet and Diestel, of turning arbitrary well-founded order trees T into graphs, in a way such that every T-branch induces a generalised path in the sense of Rado. This article contains an introduction to this method and then surveys four recent applications of order trees to infinite graphs, with relevance for well-quasi orderings, Hadwiger’s conjecture, normal spanning trees and end-structure, the last two addressing long-standing open problems by Halin.

Echo chambers in the Ising model and implications on the mean magnetization

The echo-chamber effect is a common term in opinion dynamic modeling to describe how a person’s opinion might be artificially enhanced as it is reflected back at her through social interactions. Here, we study the existence of this effect in statistical mechanics models, which are commonly used to study opinion dynamics. We show that the Ising model does not exhibit echo-chambers, but this result is a consequence of a special symmetry. We then distinguish between three types of models: (i) those with a strong echo-chamber symmetry, that have no echo-chambers at all; (ii) those with a weak echo-chamber symmetry that can exhibit echo-chambers but only if there are external fields in the system, and (iii) models without echo-chamber symmetry that generically have echo-chambers. We use these results to construct an efficient algorithm to efficiently and precisely calculate magnetization in arbitrary tree networks. Finally, we apply this algorithm to study two systems: phase transitions in the random field Ising model on a Bethe lattice and the influence optimization problem in social networks.