Ordering Of Vertices Research Articles

New concepts of inverse fuzzy mixed graphs and its application.

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between some vertices. In inverse fuzzy graph, the membership value of edges are greater than or equal to the minimum of the membership value of the corresponding vertices. In inverse fuzzy mixed graph (IFMG), directed and undirected relations exist between some vertices and it can be used to analyze many graphical problems of real life such that the membership values of edges are greater than or equal to the minimum of the membership value of the corresponding pair of vertices. In this article, the concept of IFMG is introduced first with some of its properties. Then some isomorphic properties are studied and complement of IFMG is given. Different types of operations like union, intersection, product and join between two IFMGs are defined and investigated some of their related results. An algorithm of the proposed method is executed to identify some vertices. An application is depicted using the concept IFMG to examine the order of vertices in a social network group according to communication gaps.

On Sombor index of trees

This paper is concerned with the recently introduced Sombor index SO, defined asSO=SO(G)=∑vkvℓ∈E(G)dG(vk)2+dG(vℓ)2,where dG(v) is the degree of the vertex v of a graph G. We present bounds on SO of trees in terms of order, independence number, and number of pendent vertices, and characterize the extremal cases. In addition, analogous results for quasi-trees are established.

The Referenced Vertex Ordering Problem: Theory, Applications, and Solution Methods

We introduce the referenced vertex ordering problem (REVORDER) as a combinatorial decision problem generalizing several vertex ordering problems that already appeared in the scientific literature under different guises. In other words, REVORDER is a generic problem with several possible extensions corresponding to various real-life applications. Previous works show that REVORDER has a polynomial complexity, whereas its optimization version, denoted in our work as MIN REVORDER, is NP-hard. We give a survey of methods and algorithms that can be applied to the solution of MIN REVORDER, and we develop a new enumeration scheme for its solution. Our theoretical analysis of this scheme yields several pruning techniques aimed at the reduction of the number of enumeration nodes. We then discuss how upper and lower bounds can be computed during the enumeration to design a branch-and-bound algorithm. Finally, we validate the branch-and-bound by conducting a large set of computational experiments on instances coming from various real-life applications. Our results highlight that the newly introduced pruning techniques allow for major reductions of computational cost, with a constant solution quality. In result, our branch-and-bound outperforms other existing solution methods: among 180 instances with 60 vertices, it solves 179 instances to optimality whereas the best existing method is only able to solve 109 of them. Moreover, our tests show that our algorithm can solve medium-scale instances including up to 500 vertices, which opens the perspective of handling new real-life problems. Our implementation of the branch-and-bound algorithm, together with all instances we have used, is publicly available on GitLab.

Multi-Goal Multi-Agent Path Finding via Decoupled and Integrated Goal Vertex Ordering

We introduce multi-goal multi agent path finding (MG-MAPF) which generalizes the standard discrete multi-agent path finding (MAPF) problem. While the task in MAPF is to navigate agents in an undirected graph from their starting vertices to one individual goal vertex per agent, MG-MAPF assigns each agent multiple goal vertices and the task is to visit each of them at least once. Solving MG-MAPF not only requires finding collision free paths for individual agents but also determining the order of visiting agent's goal vertices so that common objectives like the sum-of-costs are optimized.

Rendering Point Clouds with Compute Shaders and Vertex Order Optimization

AbstractIn this paper, we present several compute‐based point cloud rendering approaches that outperform the hardware pipeline by up to an order of magnitude and achieve significantly better frame times than previous compute‐based methods. Beyond basic closest‐point rendering, we also introduce a fast, high‐quality variant to reduce aliasing. We present and evaluate several variants of our proposed methods with different flavors of optimization, in order to ensure their applicability and achieve optimal performance on a range of platforms and architectures with varying support for novel GPU hardware features. During our experiments, the observed peak performance was reached rendering 796 million points (12.7GB) at rates of 62 to 64 frames per second (50 billion points per second, 802GB/s) on an RTX 3090 without the use of level‐of‐detail structures. We further introduce an optimized vertex order for point clouds to boost the efficiency of GL_POINTS by a factor of 5× in cases where hardware rendering is compulsory. We compare different orderings and show that Morton sorted buffers are faster for some viewpoints, while shuffled vertex buffers are faster in others. In contrast, combining both approaches by first sorting according to Morton‐code and shuffling the resulting sequence in batches of 128 points leads to a vertex buffer layout with high rendering performance and low sensitivity to viewpoint changes.

Computing inductive vertex orderings

Inductive vertex orderings or edge orientations of graphs, such as inductive k-independence and k-simpliciality orderings, and k-perfect orientation, arise naturally in geometric intersection graphs and have been systematically studied in Ye and Borodin (2012) [4] and Kammer and Tholey (2014) [5]. In general, those orderings and orientations are hard to compute, even approximately. In this note, we show that an inductive k-independence ordering can be computed in k-simplicial graphs, and a k-approximation (that is, a solution with approximation ratio at most k) to the optimal inductive independence ordering can be computed in k-perfectly orientable graphs. As a result, we obtain a k-approximation (2k-approximation) algorithm for maximum weight independent sets in k-simplicial (k-perfectly orientable, respectively) graphs that runs in polynomial time independent of k without prior knowledge of the ordering.

Optimizing One-time and Continuous Subgraph Queries using Worst-case Optimal Joins

We study the problem of optimizing one-time and continuous subgraph queries using the new worst-case optimal join plans. Worst-case optimal plans evaluate queries by matching one query vertex at a time using multiway intersections. The core problem in optimizing worst-case optimal plans is to pick an ordering of the query vertices to match. We make two main contributions: 1. A cost-based dynamic programming optimizer for one-time queries that (i) picks efficient query vertex orderings for worst-case optimal plans and (ii) generates hybrid plans that mix traditional binary joins with worst-case optimal style multiway intersections. In addition to our optimizer, we describe an adaptive technique that changes the query vertex orderings of the worst-case optimal subplans during query execution for more efficient query evaluation. The plan space of our one-time optimizer contains plans that are not in the plan spaces based on tree decompositions from prior work. 2. A cost-based greedy optimizer for continuous queries that builds on the delta subgraph query framework. Given a set of continuous queries, our optimizer decomposes these queries into multiple delta subgraph queries, picks a plan for each delta query, and generates a single combined plan that evaluates all of the queries. Our combined plans share computations across operators of the plans for the delta queries if the operators perform the same intersections. To increase the amount of computation shared, we describe an additional optimization that shares partial intersections across operators. Our optimizers use a new cost metric for worst-case optimal plans called intersection-cost . When generating hybrid plans, our dynamic programming optimizer for one-time queries combines intersection-cost with the cost of binary joins. We demonstrate the effectiveness of our plans, adaptive technique, and partial intersection sharing optimization through extensive experiments. Our optimizers are integrated into GraphflowDB.

General degree distance of graphs

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, the general degree distance $DD_{a,b} (G) = \sum_{ v \in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = \sum_{ w \in V(G) \setminus \{ v \} } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices.

Multi-Goal Multi-Agent Path Finding via Decoupled and Integrated Goal Vertex Ordering

We introduce multi-goal multi agent path finding (MG-MAPF) which generalizes the standard discrete multi-agent path finding (MAPF) problem. While the task in MAPF is to navigate agents in an undirected graph from their starting vertices to one individual goal vertex per agent, MG-MAPF assigns each agent multiple goal vertices and the task is to visit each of them at least once. Solving MG-MAPF not only requires finding collision free paths for individual agents but also determining the order of visiting agent's goal vertices so that common objectives like the sum-of-costs are optimized. We suggest two novel algorithms using different paradigms to address MG-MAPF: a heuristic search-based algorithm called Hamiltonian-CBS (HCBS) and a compilation-based algorithm built using the satisfiability modulo theories (SMT), called SMT-Hamiltonian-CBS (SMT-HCBS).

On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering

Given a graph G=(V,E) with a vertex ordering ≺, the fixed-order book thickness problem asks whether there is a page assignment σ such that 〈≺,σ〉 is a k-page book embedding of G. This problem is NP-complete even for any fixed k greater than 3. Recently, Bhore et al. (2019, 2020) [1,2] presented a parameterized algorithm with respect to the pathwidth κ of the vertex ordering. In this paper, we first re-analyze the running time for Bhore et al.'s algorithm, and prove a bound of 2O(κ2)⋅|V| improving Bhore et al.'s bound of κO(κ2)⋅|V|. Then, we show that fixed-order book thickness parameterized by the pathwidth of the vertex ordering does not admit a polynomial kernel unless NP ⊆ coNP/poly. Finally, we show that a generalized fixed-order book thickness problem, in which a budget of at most c crossings over all pages was given, admits a parameterized algorithm running in time (c+2)O(κ2)⋅|V|.

On the optimality of finding DMDGP symmetries

The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem, which aims to embed a weighted simple undirected graph in a Euclidean space, such that the distances between the points correspond to the values given by the weighted edges in the graph. The search space of the DMDGP is combinatorial, based on a total vertex order that implies symmetry properties related to partial reflections around planes defined by the Cartesian coordinates of three immediate and consecutive vertices that precede the so-called symmetry vertices. Since these symmetries allow us to know a priori the cardinality of the solution set and to calculate all the DMDGP solutions, given one of them, it would be relevant to identify these symmetries efficiently. Exploiting mathematical properties of the vertices associated with these symmetries, we present an optimal algorithm that finds such vertices.

Integer Programming, Constraint Programming, and Hybrid Decomposition Approaches to Discretizable Distance Geometry Problems

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in [Formula: see text] such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one, which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is on finding optimal BP trees for a class of discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting-plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low-density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem. Summary of Contribution: We present a new model to solve a combinatorial optimization problem on graphs, MIN DOUBLE, which comes from the highly active area of distance geometry and has applications in a wide variety of fields. We use integer programming (IP) and present the first constraint programming (CP) models and hybrid decomposition methods, implemented as a branch-and-cut procedure, for MIN DOUBLE. Through an extensive computational study, we show that our approaches advance the state of the art for MIN DOUBLE. We accomplish this by not only combining generic techniques from IP and CP but also exploring the structure of the problem in developing valid inequalities and variable fixing rules. Our methods significantly improve the solvability of MIN DOUBLE, which we believe can also provide insights for tackling other problem classes and applications.

On Hyper-Chordal graphs

Triangulated graphs have many interesting properties (perfection, recognition algorithms, combinatorial optimization algorithms with linear complexity). Hyper-triangulated graphs are those where each induced subgraph has a hyper-simplicial vertex. In this paper we give the characterizations of hyper-triangulated graphs using an ordering of vertices and the weak decomposition. We also offer a recognition algorithm for the hyper-triangulated graphs, the inclusions between the triangulated graphs generalizations and we show that any hyper-triangulated graph is perfect.

Cut and pendant vertices and the number of connected induced subgraphs of a graph

A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $$c>0$$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The ‘minimal’ graph in this case is a tree, thus coincides with the structure that was given by Li and Wang (Electron J Comb 19(4):P48, 2012).

An upper bound of a generalized upper Hamiltonian number of a graph

In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$-Hamiltonian number of a graph $G$. We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs $G$ and $H$ using $H$-Hamiltonian number of $G$. Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper $H$-Hamiltonian number, which is a generalization of the same claim for upper Hamiltonian numbers and upper traceable numbers. Finally we will show that for every connected graph $H$ only paths have maximal $H$-Hamiltonian number.

Feluca: A Two-Stage Graph Coloring Algorithm With Color-Centric Paradigm on GPU

There are great challenges in performing graph coloring on GPU in general. First, the long-tail problem exists in the recursion algorithm because the conflict (i.e., different threads assign the adjacent nodes to the same color) becomes more likely to occur as the number of iterations increases. Second, it is hard to parallelize the sequential spread algorithm because the color allocation depends on the adjoining iteration. Third, the atomic operation is widely used on GPU to maintain the color list, which can greatly reduce the efficiency of GPU threads. In this article, we propose a two-stage high-performance graph coloring algorithm, called Feluca, aiming to address the above challenges. Feluca combines the recursion-based method with the sequential spread-based method. In the first stage, Feluca uses a recursive routine to color a majority of vertices in the graph. Then, it switches to the sequential spread method to color the remaining vertices in order to avoid the conflicts of the recursive algorithm. Moreover, the following techniques are proposed to further improve the graph coloring performance. i) A new method is proposed to eliminate the cycles in the graph; ii) a top-down scheme is developed to avoid the atomic operation originally required for color selection; and iii) a novel color-centric coloring paradigm is designed to improve the degree of parallelism for the sequential spread part. All these newly developed techniques, together with further GPU-specific optimizations such as coalesced memory access, comprise an efficient parallel graph coloring solution in Feluca. We have conducted extensive experiments on NVIDIA GPU. The results show that Feluca can achieve 1.19 - 8.39× speedup over the state-of-the-art algorithms.

Ramsey properties and extending partial automorphisms forhbox classes of finite structures

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures.

General eccentric distance sum of graphs

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

Parameterized Algorithms for Book Embedding Problems

A $k$-page book embedding of a graph $G$ draws the vertices of $G$ on a line and the edges on $k$ half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether $G$ admits a $k$-page book embedding both when the linear order of the vertices is fixed, called ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$, or not fixed, called ${\rm B{\small OOK}~T{\small HICKNESS}}$. Both problems are known to be ${\sf NP}$-complete in general. We show that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ and ${\rm B{\small OOK}~T{\small HICKNESS}}$ are fixed-parameter tractable parameterized by the vertex cover number of the graph and that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ is fixed-parameter tractable parameterized by the pathwidth of the vertex order.

On the Minimum Size of Hamilton Saturated Hypergraphs

For $1\leq \ell< k$, an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices. A $k$-uniform hypergraph $H$ is $\ell$-hamiltonian saturated if $H$ does not contain an $\ell$-overlapping hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let sat$(N,k,\ell)$ be the smallest number of edges in an $\ell$-hamiltonian saturated $k$-uniform hypergraph on $N$ vertices. In the case of graphs Clark and Entringer showed in 1983 that sat$(N,2,1)=\lceil \tfrac{3N}2\rceil$. The present authors proved that for $k\geq 3$ and $\ell=1$, as well as for all $0.8k\leq \ell\leq k-1$, sat$(N,k,\ell)=\Theta(N^{\ell})$. Here we prove that sat$(N,2\ell,\ell)=\Theta\left(N^\ell\right)$.