Vertices Of Graph Research Articles

Spanning trees for many different numbers of leaves

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.

Distance Degree Sequence and Distance Neighborhood Degree Sequence of Vertices in Fuzzy Graphs and Some of Their Properties

Objectives: The objective of this study is to explore new types of sequences in fuzzy graphs namely distance degree sequence and distance neighborhood degree sequence of fuzzy graphs which may have various applications in the real world. Methods: In this paper, two unique sequences for each vertex of fuzzy graphs are given by means of its degree and neighborhood degree which are called as the distance degree sequence and distance neighborhood degree sequence of vertices of fuzzy graphs respectively. The n-tuple of the distance degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance degree sequence of the fuzzy graph and the n-tuple of distance neighborhood degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance neighborhood degree sequence of the fuzzy graph. Findings: The distance degree sequence and distance neighborhood degree sequence of fuzzy graphs are introduced. The properties of these sequences and the general forms of these sequences of various fuzzy graphs are given. The relationship between the distance degree sequence and distance neighborhood degree sequences is discussed. Novelty: The distance degree sequence of graphs has already been introduced. In fuzzy graph theory, the distance degree sequence and distance neighborhood degree sequence are introduced in this paper. Mathematics Subject Classification (202 0): 03E72, 05C72, 05C12. Keywords: Fuzzy graphs, Distance degree sequence, Distance neighborhood-degree sequence, Regular fuzzy graphs

On the automorphisms of the Drinfeld modular groups

Let [Formula: see text] be the ring of elements in an algebraic function field [Formula: see text] over [Formula: see text] which are integral outside a fixed place [Formula: see text]. In contrast to the classical modular group [Formula: see text] and the Bianchi groups, the Drinfeld modular group [Formula: see text] is not finitely generated and its automorphism group [Formula: see text] is uncountable. Except for the simplest case [Formula: see text] not much is known about the generators of [Formula: see text] or even its structure. We find a set of generators of [Formula: see text] for a new case. On the way, we show that every automorphism of [Formula: see text] acts on both the cusps and the elliptic points of [Formula: see text]. Generalizing a result of Reiner for [Formula: see text] we describe for each cusp an uncountable subgroup of [Formula: see text] whose action on [Formula: see text] is essentially defined on the stabilizer of that cusp. In the case where [Formula: see text] (the degree of [Formula: see text]) is [Formula: see text], the elliptic points are related to the isolated vertices of the quotient graph [Formula: see text] of the Bruhat–Tits tree. We construct an infinite group of automorphisms of [Formula: see text] which fully permutes the isolated vertices with cyclic stabilizer.

Polynomial Removal Lemma for Ordered Matchings

We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.

Pipe production cost management model based on graph theory

The pipe production process is a complex dynamical system with a significant number of production operations and interconnected cost localization centres. Purpose. To develop a mathematical model of the process of handling analytical information of a pipe enterprise using graph theory for the needs of cost management at each production stage. Methodology. Technological flow charts of the production job, where the list of mandatory activities and routing of the operating process are indicated, were used to organize cost accounting by operational centres of their localization with the help of computer modelling elements, namely the graph theory. The work used a complex of research methods, including analysis and scientific synthesis of scientific and technical information; theoretical studies; methods of mathematical and computer modelling, engineering developments. Findings. The cost accounting process is represented by a directed hypergraph. Each production operation is associated with a graph vertex. Each vertex is assigned to a series of interrelated marks: type of technological operation; weight factor; the value of direct costs for the operation; the value of indirect costs for the operation; the expense factor. The initial state of the hypergraph is specified by means of its initial marking, which corresponds to the set cipher of the batch of pipes and its weight. Further execution of the marked hypergraph is performed by running the allowed vertices. Marking of the graph ends if all the information from the technological flow charts of pipe products has been used. Originality. A model of cost management by operational centres is proposed which will allow dividing the prime cost of pipes which differ according to the production technology. It is recommended to rank pipe products within each technologically similar group for an economically feasible allocation of costs between batches of pipes. The proposed process model of processing analytical information of the pipe enterprise using graph theory will allow providing data governance as for the prime cost of each individual batch of pipe products, will contribute to the improvement in the order generation procedure taking into account the acquired information, and will allow making the appropriate adjustments when making management decisions. Practical value. Calculation of expenses for production operations allows one to identify the most cost-intensive of them. Relying on the results of the calculation, the enterprise management has the opportunity to manage costs at any stage of the production process. Moreover, the model allows determining the amount of consumed metal for each production operation, which is relevant for pipe enterprises. It is appropriate to focus further scientific developments in this direction on the possibility of managing non-manufacturing cost with the help of mathematical models in economics to optimize the supply and sales activities of the enterprise.

Exact and parameterized algorithms for the independent cutset problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is ▪-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a O⁎(1.4423n)-time algorithm to compute a minimum independent cutset (if any). Since the property of having an independent cutset is MSO1-expressible, our main results are concerned with structural parameterizations for the problem considering parameters incomparable with clique-width. We present ▪-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to P5-free graphs. We close by introducing the notion of α-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.

Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

Adaptive collective responses to local stimuli in anonymous dynamic networks

We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents are represented by vertices in a dynamic graph G whose edge set changes over time, and stimuli are placed adversarially on the vertices of G where each agent is only capable of recognizing a co-located stimulus. Agents communicate via token passing along edges to alert other agents to transition to an Aware state when stimuli are present and an Unaware state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that can handle arbitrary adversarial stimulus dynamics, while an adversary (or the agents themselves) reconfigures the connections (edges) of G over time in a controlled way. This algorithm can be used to solve the foraging problem on reconfigurable graphs where, in addition to food sources (stimuli) being discovered, removed, or shifted arbitrarily, we would like the agents to consistently self-organize, using only local interactions, such that if the food remains in a position long enough, the agents transition to a gather phase in which many collectively form a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should undergo a self-induced collective phase change and switch to a search phase in which they distribute themselves randomly throughout the graph to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable, withstanding competing broadcast waves of state transition that may interfere with each other. Like a physical phase change, such as the ferromagnetic models underlying the gather and search algorithms used for foraging, microscopic changes in the environment trigger these macroscopic, system-wide transitions as agents share information and respond locally to get the desired collective response.

A localized approach for Turán number of long cycles

AbstractLet be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by over all edges in an ‐vertex graph is at most , and characterize all ‐vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

GABoost: Graph Alignment Boosting via Local Optimum Escape

Heterogeneous graphs provide a universal data structure for representing various kinds of structured data in numerous domains. The graph alignment problem aims to find the correspondences of vertices in different graphs, playing a fundamental role in many downstream tasks of heterogeneous graph mining. In recent years, many graph alignment methods have been proposed, ranging from classical optimization methods , spectral methods , to embedding learning based-methods . Due to the problem's complexity, the result found by most existing methods is either a heuristic solution or a critical point in the solution space. In this paper, we propose GABoost, a graph alignment boosting algorithm that takes as input an initial alignment between two heterogeneous graphs and outputs a boosted alignment via an iterative local-optimum-escape process. One of the distinctive features of GABoost is that it can be sequentially composed with any graph alignment methods to improve the output of upstream methods. To examine the effectiveness of GABoost, we select 7 upstream methods of graph alignment as well as 6 real-world datasets, and quantitatively investigate the degree to which GABoost boosts these methods. The results show that GABoost improves the alignment accuracy of the 7 upstream methods by 25.25% on average with acceptable time overhead.

An Alternate Form of Merino-Mǐcka-Mütze’s Solution to a Knuth’s Combinatorial Generation Problem

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

Pancyclicity of Hamiltonian graphs

An n -vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from 3 up to n . In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most k and at least n = \Omega(k^{2}) vertices is pancyclic. We prove this old conjecture in a strong form by showing that if such a graph has n = (2+o(1))k^{2} vertices, it is already pancyclic, and this bound is asymptotically best possible.

Sticky Brownian motions on star graphs

AbstractThis paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to the previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure $$\varPhi $$ Φ . The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure $$\varPhi $$ Φ . Extensions to general graph structures can be given by applying to our results a localisation technique.

Compatible powers of Hamilton cycles in dense graphs

AbstractThe notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.

Prokrustean Graph: A substring index for rapid k-mer size analysis.

Despite the widespread adoption of -mer-based methods in bioinformatics, understanding the influence of -mer sizes remains a persistent challenge. Selecting an optimal -mer size or employing multiple -mer sizes is often arbitrary, application-specific, and fraught with computational complexities. Typically, the influence of -mer size is obscured by the outputs of complex bioinformatics tasks, such as genome analysis, comparison, assembly, alignment, and error correction. However, it is frequently overlooked that every method is built above a well-defined -mer-based object like Jaccard Similarity, de Bruijn graphs, -mer spectra, and Bray-Curtis Dissimilarity. Despite these objects offering a clearer perspective on the role of -mer sizes, the dynamics of -mer-based objects with respect to -mer sizes remain surprisingly elusive. This paper introduces a computational framework that generalizes the transition of -mer-based objects across -mer sizes, utilizing a novel substring index, the Prokrustean graph. The primary contribution of this framework is to compute quantities associated with -mer-based objects for all -mer sizes, where the computational complexity depends solely on the number of maximal repeats and is independent of the range of -mer sizes. For example, counting vertices of compacted de Bruijn graphs for can be accomplished in mere seconds with our substring index constructed on a gigabase-sized read set. Additionally, we derive a space-efficient algorithm to extract the Prokrustean graph from the Burrows-Wheeler Transform. It becomes evident that modern substring indices, mostly based on longest common prefixes of suffix arrays, inherently face difficulties at exploring varying -mer sizes due to their limitations at grouping co-occurring substrings. We have implemented four applications that utilize quantities critical in modern pangenomics and metagenomics. The code for these applications and the construction algorithm is available at https://github.com/KoslickiLab/prokrustean.

On Null Vertex in Bipolar Fuzzy Graphs

We present a novel vertex in Bipolar fuzzy graph, null vertex, which is distinct from boundary vertex and interior vertex and also attempt a study on null vertex in bipolar fuzzy closed helm graph CHn.

GKE-TUNet: Geometry-Knowledge Embedded TransUNet Model for Retinal Vessel Segmentation Considering Anatomical Topology.

Automated retinal vessel segmentation is crucial for computer-aided clinical diagnosis and retinopathy screening. However, deep learning faces challenges in extracting complex intertwined structures and subtle small vessels from densely vascularized regions. To address these issues, we propose a novel segmentation model, called Geometry-Knowledge Embedded TransUNet (GKE-TUNet), which incorporates explicit embedding of topological features of retinal vessel anatomy. In the proposed GKE-TUNet model, a skeleton extraction network is pre-trained to extract the anatomical topology of retinal vessels from refined segmentation labels. During vessel segmentation, the dense skeleton graph is sampled as a graph of key-points and connections and is incorporated into the skip connection layer of TransUNet. The graph vertices are used as node features and correspond to positions in the low-level feature maps. The graph attention network (GAT) is used as the graph convolution backbone network to capture the shape semantics of vessels and the interaction of key locations along the topological direction. Finally, the node features obtained by graph convolution are read out as a sparse feature map based on their corresponding spatial coordinates. To address the problem of sparse feature maps, we employ convolution operators to fuse sparse feature maps with low-level dense feature maps. This fusion is weighted and connected to deep feature maps. Experimental results on the DRIVE, CHASE-DB1, and STARE datasets demonstrate the competitiveness of our proposed method compared to existing ones.

Algorithms for subgraph complementation to some classes of graphs

For a class G of graphs, the objective of Subgraph Complementation toG is to find whether there exists a subset S of vertices of the input graph G such that modifying G by complementing the subgraph induced by S results in a graph in G. We obtain a polynomial-time algorithm for the problem when G is the class of graphs with minimum degree at least k, for a constant k, answering an open problem by Fomin et al. (Algorithmica, 2020). When G is the class of graphs without any induced copies of the star graph on t+1 vertices (for any constant t≥3) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when G is the class of graphs without any induced copies of the star graph on t+1 vertices, for every constant t≥5.