Cycles In Graphs Research Articles

Cyclic Action Graphs for goal recognition problems with inaccurately initialised fluents

Goal recognisers attempt to infer an agent’s intentions from a sequence of observed actions. This is an important component of intelligent systems that aim to assist or thwart actors; however, there are many challenges to overcome. For example, the initial state of the environment could be partially unknown, and agents can act suboptimally and observations could be missing. Approaches that adapt classical planning techniques to goal recognition have previously been proposed, but, generally, they assume the initial world state is accurately defined. In this paper, a state is inaccurate if any fluent’s value is unknown or incorrect. Our aim is to develop a goal recognition approach that is as accurate as the current state-of-the-art algorithms and whose accuracy does not deteriorate when the initial state is inaccurately defined. To cope with this complication, we propose solving goal recognition problems by means of an Action Graph. An Action Graph models the dependencies, i.e. order constraints, between all actions rather than just actions within a plan. Leaf nodes correspond to actions and are connected to their dependencies via operator nodes. After generating an Action Graph, the graph’s nodes are labelled with their distance from each hypothesis goal. This distance is based on the number and type of nodes traversed to reach the node in question from an action node that results in the goal state being reached. For each observation, the goal probabilities are then updated based on either the distance the observed action’s node is from each goal or the change in distance. Our experimental results, for 15 different domains, demonstrate that our approach is robust to inaccuracies within the defined initial state.

Group-supermagic labeling of Cartesian products of two even cycles

A Γ-supermagic labeling of a graph G=(V,E) with |E|=q is a bijection from E to an Abelian group Γ of order q such that the sum of labels of all incident edges of every vertex x∈V is equal to the same element μ∈Γ. An existence of a Γ-supermagic labeling of Cartesian product of two cycles Cm□Cn for every m,n≥3 by Z2mn and for any two odd cycles Cm□Cn for any m,n≥3 and any Abelian group Γ of order 2mn was proved recently. In this paper we completely solve the case of m,n both even and present a labeling method for all Abelian groups of order 2mn, where m,n are even and greater than two.

Lutosławski's Mid-Century Harmony

Abstract This article proposes to expand our understanding of Witold Lutosławski's chordal space and its relation to formal processes. It argues that the traditional analytical focus on harmonic colors of isolated chords comes at the expense of understanding how chordal interactions shape form. How do individual chords participate in cogent, expressive, and sensuous long-range harmonic processes? The article explores how the arrangement of intervals in twelve-tone chords (especially the ordering, restriction, and modularity of intervals) not only imprint distinct coloristic qualities to single chordal entities but are also implicated in strategies of harmonic progression, relatedness, complementarity, and contrast. These analytical goals prompt a reconceptualization of chordal elements, which are understood as combinations of pitch class and modal quality rather than simply as pitches in register. The analytical focus is on Lutosławski's harmony of the mid- to late fifties, attending in particular to the Five Iłłakowicz Songs, the composer's inaugural effort in the use of vertical aggregates. Lutosławski's chordal space in these pieces is interpreted through a series of cyclic graphs and transpositional networks, prompting a novel understanding between twelve-tone chordal harmony and form. The study proposes to contribute to the analytical reappraisal of a set of pieces that anchor an influential harmonic practice in twentieth- and twenty-first-century music.

Decentralized formation of multi‐agent conformable fractional nonlinear robot systems

AbstractA novel approach to address the asymptotic stability problem of the decentralized formation control of multi‐agent robot systems considering a cyclic pursuit formation graph is derived by using the classical Lyapunov's stability method and LaSalle's invariance principle. Theoretical development of a local potential function‐based control law and the formulation of the position and formation error dynamics, from the viewpoint of the multi‐variable conformable fractional‐order calculus, is given. In addition, a comparison between the integer‐order case and conformable fractional‐order case is provided. Finally, a numerical example illustrates the performance of the developed results.

On the monoid of partial isometries of a cycle graph

On the monoid of partial isometries of a cycle graph

Streaming deletion problems parameterized by vertex cover

Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of Π-free Deletion, H-free Deletion, and the more specific forms of Cluster Vertex Deletion and Odd Cycle Transversal. We focus on the complexity in terms of the number of passes over the input stream, and the memory used. This leads to a pass/memory trade-off, where a different algorithm might be favourable depending on the context and instance. We also discuss implications for parameterized complexity in the non-streaming setting.

On the spectra of token graphs of cycles and other graphs

The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn).

Partition Dimention of Amalgamation-Side in Cycle Graph

The graph is a pair of sets , where is a finite set whose elements are called vertices, and is the set of pairs of members of . which is called the edge. Let be a simple graph where . The distance between points and is denoted by is the length of the shortest path between and . Given and there is a vertex on the connected graph , then the distance between and is denoted . If is -partition of , then the representation of with respect to is -ordered pairs, . If the -ordered pairs for are all different, then the partition is called a dimension partition. The minimal -number which is the -differentiating partition of is called the partition dimension of and is denoted by . In this study, the partition dimensions of the sided amalgamation result will be determined on an even-order cycle graph. In determining the dimensions of the partition, characterization of the partition dimensions is used in the path graph, the lemma about the distinguishing set and the equivalence point, especially in the even-order cycle graph. The results of this study are pd(Amal(Cn,e,k)) = 3 for n≥4 , pd(Amal(C4,e,k))=4 for k=4 , pd(Amal(C4,e,k))=3+m for k=2m+3 and k=2m+4 where m=1,2,3,...

Quotient of quotient graph a novel approach to compute π‐conjugated dendrimer and predict its properties

AbstractThe role of dendrimers is getting updated and grabbing immense attention in many fields. Dendrimers are used in a broad spectrum, in specific, it is widely utilized in drug delivery, for targeted carrier and specific action. Also, in drug catalysis, it improves the solubility of poorly soluble drugs and increases the stability of active ingredients within the cores. Many researchers are interested in developing new dendrimers by using various compounds. ‐conjugated dendrimers via truxenes and thienylethynylene units, are considered a promising material in the field of medicinal chemistry, bio‐organic, and environmental sciences. In QSAR/QSPR, the structural analysis of chemical compounds is examined by topological descriptors. In this article, the various distance‐based topological indices such as Wiener, Szeged, and Mostar of gradient star‐persistent ‐conjugated dendrimers are determined. Since gradient star‐persistent ‐conjugated dendrimers have odd cycles, the usual cut method is not applicable, therefore we use a new concept, the quotient of quotient graph to determine the numerical expression of various indices. An analysis of linear regression reveals that degree‐based topological descriptors predict better physicochemical properties.

Direct co-deposition of binder-free Cu-biochar-based nonenzymatic disposable sensing element for electrochemical glucose detection

This work is the first attempt to incorporate biochar on a pencil graphite electrode via one-step electrodeposition as a rational route for refining a copper-based biosensor for a high-performance nonenzymatic electrochemical glucose sensor. Biochar undecorated electrode counterpart was also fabricated for comparison. The cyclic voltammetry graph showed that co-deposition with biochar enhanced the electrocatalytic activity toward glucose oxidation in an alkaline medium. The surface morphology of the copper-biochar hybrid coating demonstrated an attractive three-dimensional structure that had not been reported in the literature. The results indicated that the supplied extending surface area due to biochar modification was the controlling factor for the improved electrochemical reactivity for glucose sensing. The amperometric analysis at an optimum potential of +0.70 V showed linear detection ranges from 0.8 µM to 1 mM and from 1 mM to 5 mM, and good characteristics, such as high sensitivity (6214.4 µA mM−1 cm–2), low limit of detection (0.8 µM), a fast response time (less than 2 s), and excellent anti-interference capability toward the oxidation of glucose. The cost-effective and facile fabrication suggests that the prepared glucose-sensing element can be potentially adapted for construction as a disposable.

Super oriented cycles in permutations

The degree of dissimilarity between genome sequences of homologous species is a measure of the evolutionary distance between them. It serves as a metric in the construction of phylogenetic trees, which depict the evolutionary relationships and common ancestry among different species. Given two genome sequences, evolutionary distance is determined by estimating the number of global mutations that transform one sequence to the other. The computation of the evolutionary distance is done by modelling a genome with the corresponding permutation. Global rearrangement operations such as transposition that model a particular genomic mutation are studied by employing a combinatorial structure known as a cycle graph of the corresponding permutation. A cycle in a cycle graph that has odd length is called an odd cycle. In the context of the problem of sorting by transpositions (SBT), a valid 2-move is a transposition that increases the number of odd cycles in the cycle graph by two. A super oriented cycle (SOC) is an odd cycle C where C and one of the resultant cycles admit valid 2-moves. The minimum number of mutations required to transform a species S into a related species T is the distance from S to T under that mutation. Christie opined that characterizing SOCs will improve the lower bound of the transposition distance. We characterize super oriented cycles. Equivalent transformations on permutations like reduction and (g,b)-split preserve the transposition distance of a given permutation and map SBT to the corresponding SBT on a transformed simpler permutation. We introduce merge, a novel equivalent transformation. These results have applications in computing transposition and other distances between related species.

Accuracy analysis for distributed dynamic state estimation in large-scale systems with a cyclic network graph

Accuracy analysis for distributed dynamic state estimation in large-scale systems with a cyclic network graph

Recurrent generation of maximally entangled single-particle states via quantum walks on cyclic graphs

Recurrent generation of maximally entangled single-particle states via quantum walks on cyclic graphs

Brooks' Theorem in Graph Streams: A Single-Pass Semi-Streaming Algorithm for $\Delta$-Coloring

Every graph with maximum degree $\Delta$ can be colored with $(\Delta+1)$ colors using a simple greedy algorithm. Remarkably, recent work has shown that one can find such a coloring even in the semi-streaming model. But, in reality, one almost never needs $(\Delta+1)$ colors to properly color a graph. Indeed, the celebrated \Brooks' theorem states that every (connected) graph beside cliques and odd cycles can be colored with $\Delta$ colors. Can we find a $\Delta$-coloring in the semi-streaming model as well? We settle this key question in the affirmative by designing a randomized semi-streaming algorithm that given any graph, with high probability, either correctly declares that the graph is not $\Delta$-colorable or outputs a $\Delta$-coloring of the graph. The proof of this result starts with a detour. We first (provably) identify the extent to which the previous approaches for streaming coloring fail for $\Delta$-coloring: for instance, all these approaches can handle streams with repeated edges and they can run in $o(n^2)$ time -- we prove that neither of these tasks is possible for $\Delta$-coloring. These impossibility results however pinpoint exactly what is missing from prior approaches when it comes to $\Delta$-coloring. We then build on these insights to design a semi-streaming algorithm that uses $(i)$ a novel sparse-recovery approach based on sparse-dense decompositions to (partially) recover the "problematic" subgraphs of the input -- the ones that form the basis of our impossibility results -- and $(ii)$ a new coloring approach for these subgraphs that allows for recoloring of other vertices in a controlled way without relying on local explorations or finding "augmenting paths" that are generally impossible for semi-streaming algorithms. We believe both these techniques can be of independent interest.

Erratum. Dominator coloring of total graph of path and cycles

Erratum. Dominator coloring of total graph of path and cycles

Di-Forcing Polynomials for Cyclic Ladder Graphs CLn

The cyclic ladder graph CLn is the Cartesian product of cycles Cn and paths P2, that is CLn=Cn×P2, (n≥3). The di-forcing polynomial of CLn is a binary enumerative polynomial of all perfect matching forcing and anti-forcing numbers. In this paper, we derive recursive formulas for the di-forcing polynomial of cyclic ladder graph CLn by classifying and counting the matching cases of the associated edges of a given vertex, from which we obtain the number of perfect matching, the forcing and anti-forcing polynomials, and the generating function and by computing some di-forcing polynomials of the lower order CLn.

Patterns and Line-Adjacency Matrix Strategy on a Game of Induced Matching

A matching is an independent set of edges in a graph G; an induced matching is a matching with an additional property that no two of its edges are joined by an edge in G. An induced matching M in a graph G is maximal if no other induced matching in G contains M. In the proposed game, players alternate choose edges on a graph while maintaining an induced matching. They continue until the matching is maximal (i.e. no player can choose another edge), and the last player to choose an edge wins. This paper will discuss patterns for this game and prove results on several categories of graphs including Complete Graphs, Path Graphs, Cycle Graphs, and Ladder Graphs. In addition, a novel Line-Adjacency Matrix method has been defined and proved to calculate all possible edges for the next move in a connected graph. Keywords: Induced Matching, Adjacency Matrix, Line-Adjacency Graph

The Turán Density of Tight Cycles in Three-Uniform Hypergraphs

Abstract The Turán density of an $r$-uniform hypergraph ${\mathcal {H}}$, denoted $\pi ({\mathcal {H}})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of ${\mathcal {H}}$, as $n \to \infty $. Denote by ${\mathcal {C}}_{\ell }$ the $3$-uniform tight cycle on $\ell $ vertices. Mubayi and Rödl gave an “iterated blow-up” construction showing that the Turán density of ${\mathcal {C}}_{5}$ is at least $2\sqrt {3} - 3 \approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain ${\mathcal {C}}_{\ell }$ for larger $\ell $ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of ${\mathcal {C}}_{\ell }$ for all large $\ell $ not divisible by $3$, showing that indeed $\pi ({\mathcal {C}}_{\ell }) = 2\sqrt {3} - 3$. To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement “a graph is bipartite if and only if it does not contain an odd cycle”.

Cycle Existence for All Edges in Folded Hypercubes under Scope Faults

The reliability of large-scale networks can be compromised by various factors such as natural disasters, human-induced incidents such as hacker attacks, bomb attacks, or even meteorite impacts, which can lead to failures in scope of processors or links. Therefore, ensuring fault tolerance in the interconnection network is vital to maintaining system reliability. The n-dimensional folded hypercube network structure, denoted as FQn, is constructed by adding an edge between every pair of vertices with complementary addresses from an n-dimensional hypercube, Qn. Notably, FQn exhibits distinct characteristics based on the dimensionality: it is bipartite for odd integers n≥3 and non-bipartite for even integers n≥2. Recently, in terms of the issue of how FQn performs in communication under regional or widespread destruction, we mentioned that in FQn, even when a pair of adjacent vertices encounter errors, any fault-free edge can still be embedded in cycles of various lengths. Additionally, even when the smallest communication ring experiences errors, it is still possible to embed cycles of any length. The smallest communication ring in FQn is observed to be the four-cycle ring. In order to further investigate the communication capabilities of FQn, we further discuss whether every fault-free edge will still be a part of every communication ring with different lengths when the smallest communication ring is compromised in FQn. In this study, we consider a fault-free edge e=(u,v) and F4={f1, f2,f3, f4} as the set of faulty extreme vertices for any four cycles in FQn. Our research focuses on investigating the cycle-embedding properties in FQn−F4, where the fault-free edges play a significant role. The following properties are demonstrated: (1) For n≥4 in FQn−F4, every even length cycle with a length ranging from 4 to 2n−4 contains a fault-free edge e; (2) For every even n≥4 in FQn−F4, every odd length cycle with a length ranging from n+1 to 2n−5 contains a fault-free edge e. These findings provide insights into the cycle-embedding capabilities of FQn, specifically in the context of fault tolerance when considering certain sets of faulty vertices.

Formal Verification of Sequential Circuits in Superconducting Single Flux Quantum Technologies

Logical equivalence checking is one of the most effective verification steps in the entire design process of integrated circuits. However, it faces challenges in emerging technologies due to differences between its logic models and those of the standard complementary metal oxide semiconductor (CMOS). This paper presents qSC, an equivalence checking framework targeting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sequential</i> circuits mapped to rapid superconducting single-flux-quantum (RSFQ) logic circuits. In addition to a logical checking module, qSC also includes several structural checking modules. The structural checking modules are used to check whether the circuit meets the design rules of superconducting RSFQ logic circuits. The main difficulty of verifying nonlinear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sequential</i> circuits is in finding the levels of logic cells in the circuit for full-path-balancing checking for which the conventional topological sorting algorithm of acyclic digraphs is not applicable. The key is to define and calculate levels for nodes in a cyclic graph and to avoid any loop trap. Hence, a new algorithm is presented for full-path-balancing checking of the cyclic digraph that satisfies all of these desirable features. The experimental results show the feasibility of qSC on sequential circuits implemented in RSFQ technologies.