Ordering Of Vertices Research Articles

On the extremal values for the Mostar index of trees with given degree sequence

For a given graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between nu(e) and nv(e) over all edges e=uv of G, where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. The degree sequence of a tree is the sequence of the degrees (in descending order) of the non-leaf vertices. This paper determines those trees with a given degree sequence which have the greatest Mostar index. Consequently, all extremal trees with the greatest Mostar index are obtained in the sets of all trees of order n with the maximum degree, the number of leaves, the independence number and the matching number, respectively. On the other hand, some properties of trees with a given degree sequence which have the least Mostar index are given. We also determine those trees with exactly n−3 or n−4 leaves when the degree sequence is given, which have the least Mostar index. At last some numerical results are discussed, in which we calculate the Mostar indices of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons; We compare their Mostar indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Mostar index is better than or as good as the models corresponding to the other distance-based indices.

Coloring squares of graphs via vertex orderings

We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that [Formula: see text] is [Formula: see text]-colorable when [Formula: see text] is a cocomparability graph, [Formula: see text]-colorable when [Formula: see text] is a strongly orderable graph and [Formula: see text]-colorable when [Formula: see text] is a dually chordal graph, where [Formula: see text] is the maximum degree and [Formula: see text] = max[Formula: see text] is the multiplicity of the graph [Formula: see text]. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.

Drug-target interaction prediction using semi-bipartite graph model and deep learning

BackgroundIdentifying drug-target interaction is a key element in drug discovery. In silico prediction of drug-target interaction can speed up the process of identifying unknown interactions between drugs and target proteins. In recent studies, handcrafted features, similarity metrics and machine learning methods have been proposed for predicting drug-target interactions. However, these methods cannot fully learn the underlying relations between drugs and targets. In this paper, we propose anew framework for drug-target interaction prediction that learns latent features from drug-target interaction network.ResultsWe present a framework to utilize the network topology and identify interacting and non-interacting drug-target pairs. We model the problem as a semi-bipartite graph in which we are able to use drug-drug and protein-protein similarity in a drug-protein network. We have then used a graph labeling method for vertex ordering in our graph embedding process. Finally, we employed deep neural network to learn the complex pattern of interacting pairs from embedded graphs. We show our approach is able to learn sophisticated drug-target topological features and outperforms other state-of-the-art approaches.ConclusionsThe proposed learning model on semi-bipartite graph model, can integrate drug-drug and protein-protein similarities which are semantically different than drug-protein information in a drug-target interaction network. We show our model can determine interaction likelihood for each drug-target pair and outperform other heuristics.

Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as in the sparse graph regime (with the average degree smaller than ) and in the dense graph regime, for a small positive constant . Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.

On compatible triangulations with a minimum number of Steiner points

Two vertex-labelled polygons are compatible if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations—for every face, the clockwise cyclic order of vertices on the boundary must be the same. It is known that every pair of compatible n-vertex polygonal regions can be extended to compatible triangulations by adding O(n2) Steiner points. Furthermore, Ω(n2) Steiner points are sometimes necessary, even for a pair of polygons. Compatible triangulations provide piecewise linear homeomorphisms and are also a crucial first step in morphing planar graph drawings, aka “2D shape animation.” An intriguing open question, first posed by Aronov, Seidel, and Souvaine in 1993, is to decide if two compatible polygons have compatible triangulations with at most k Steiner points. In this paper we prove the problem to be NP-hard for polygons with holes. The question remains open for simple polygons.

A novel approach to generating microstructurally-aware non-convex domains

We present a novel methodology for generating synthetic microstructures in non-convex domains. Simple techniques that include mapping, point identification, divide and map, clockwise and counter-clockwise ordering of vertices, approximate convex decomposition and stitching are proposed to realize tessellated non-convex geometries, both in two- and three-dimensions. A tessellated convex domain is used as a reference for mapping of either simply or multiply connected non-convex geometries that are difficult to generate through simple cut operations using geometric primitives. Subroutines are developed in MATLAB for mapping the tessellations. The developed routines are also capable of generating geometry files that allow in further discretization or meshing of the tessellated domain. We present several case studies in two- and three-dimensions to demonstrate the robustness of the tool developed and highlight the unique capabilities of the proposed methodology in handling non-convex internal and external boundaries of the domain to be mapped.

Learning random points from geometric graphs or orderings

Let Xv for v∈V be a family of n iid uniform points in the square . Suppose first that we are given the random geometric graph , where vertices u and v are adjacent when the Euclidean distance dE(Xu,Xv) is at most r. Let n3/14≪r≪n1/2. Given G (without geometric information), in polynomial time we can with high probability approximately reconstruct the hidden embedding, in the sense that “up to symmetries,” for each vertex v we find a point within distance about r of Xv; that is, we find an embedding with “displacement” at most about r. Now suppose that, instead of G we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v. Then, with high probability, in polynomial time we can find an embedding with displacement .

Distance spectral radius of trees with given number of segments

A segment of a tree is a path-subtree whose terminal vertices are branching or pendant vertices. We determine the structure of trees that maximize the distance spectral radius among trees with given order and number of segments, and among trees with given order and number of vertices of degree 2, respectively.

General eccentric connectivity index of trees and unicyclic graphs

We introduce the general eccentric connectivity index of a graph G, ECIα(G)=∑v∈V(G)eccG(v)dGα(v) for α∈R, where V(G) is the vertex set of G, eccG(v) is the eccentricity of a vertex v and dG(v) is the degree of v in G. We present lower and upper bounds on the general eccentric connectivity index for trees of given order, trees of given order and diameter, and trees of given order and number of pendant vertices. Then we give lower and upper bounds on the general eccentric connectivity index for unicyclic graphs of given order, and unicyclic graphs of given order and girth. The upper bounds for trees of given order and diameter, and trees of given order and number of pendant vertices hold for α>1. All the other bounds are valid for 0<α≤1 or 0<α<1. We present all the extremal graphs, which means that our bounds are best possible.

Exploratory Combinatorial Optimization with Reinforcement Learning

Many real-world problems can be reduced to combinatorial optimization on a graph, where the subset or ordering of vertices that maximize some objective function must be found. With such tasks often NP-hard and analytically intractable, reinforcement learning (RL) has shown promise as a framework with which efficient heuristic methods to tackle these problems can be learned. Previous works construct the solution subset incrementally, adding one element at a time, however, the irreversible nature of this approach prevents the agent from revising its earlier decisions, which may be necessary given the complexity of the optimization task. We instead propose that the agent should seek to continuously improve the solution by learning to explore at test time. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in principle, applicable to any combinatorial problem that can be defined on a graph. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. Moreover, because ECO-DQN can start from any arbitrary configuration, it can be combined with other search methods to further improve performance, which we demonstrate using a simple random search.

General Multiplicative Zagreb Indices of Graphs with a Small Number of Cycles

We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results.

Testing a Heuristic Algorithm for Finding a Maximum Clique on DIMACS and Facebook Graphs

In this paper we propose a new heuristic algorithm for solving a maximum clique search problem (MCP). While the proposed algorithm (called TrustCLQ) uses a general approach to solving MCP, it is almost independent of the order of vertices and does not exploit a partition of the graph into independent sets. The algorithm was tested on DIMACS library graphs which are often employed for testing MCP solution algorithms. TrustCLQ algorithm was compared with the well-known ILS heuristic algorithm (as well as with a standard algorithm from networkx library) on DIMACS data sets. Moreover, TrustCLQ algorithm has been tested on Facebook social graphs

Bipartite Analogues of Comparability and Cocomparability Graphs

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 24 May 2019Accepted: 30 June 2020Published online: 15 September 2020Keywordscocomparability bigraph, chordal bigraph, interval bigraph, interval containment bigraph, two-directional orthogonal-ray graph, characterization, orientation, vertex ordering, invertible pair, asteroid, edge-asteroid, recognition, polynomial time algorithmAMS Subject Headings05C75, 05C85Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Quasi-Tree Graphs With Extremal General Multiplicative Zagreb Indices

Zagreb indices and their modified versions of a molecular graph are important molecular descriptors which can be applied in characterizing the structural properties of organic compounds from different aspects. In this article, by exploring the structures of the quasi-tree graphs with given different parameters (order, perfect matching and number of pendant vertices) and using the properties of the general multiplicative Zagreb indices, we determine the minimal and maximal values of general multiplicative Zagreb indices on quasi-tree graphs with given order, with perfect matchings, and with given number of pendant vertices. Furthermore, we present the minimal and maximal values of general multiplicative Zagreb indices on trees with perfect matchings.

NK-MaxClique and MMCQ: Tow New Exact Branch and Bound Algorithms for the Maximum Clique Problem

The maximum clique problem (MCP) is a fundamental problem in combinatorial optimization which finds important applications in real-word. This paper describes two new efficient branch-and-bound maximum clique algorithms NK-MaxClique and MMCQ, designed for solving MCP. We define some pruning conditions based on core numbers and vertex ordering to efficiently remove many of the search space. With respect to this ordering, the algorithms consider the vertices respectively to find the corresponding maximum clique in subproblems. Simulation results demonstrate that the algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs.

Vertex ordering with optimal number of adjacent predecessors

In this paper, we study the complexity of the selection of a graph discretization order with a stepwise linear cost function. Finding such vertex ordering has been proved to be an essential step to solve discretizable distance geometry problems (DDGPs). DDGPs constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. In particular, it is useful to find discretization orders that minimize an indicator of the size of the search tree. Our stepwise linear cost function generalizes this situation and allows to discriminate the vertices into three categories depending on the number of adjacent predecessors of each vertex in the order and on two parameters K and U. We provide a complete study of NP-completeness for fixed values of K and U. Our main result is that the problem is NP-complete in general for all values of K and U such that U ≥ K + 1 and U ≥ 2. A consequence of this result is that the minimization of vertices with exactly K adjacent predecessors in a discretization order is also NP-complete.

A recognition algorithm for simple-triangle graphs

A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a longstanding open problem, which was recently settled. This paper provides a new recognition algorithm for simple-triangle graphs to improve the time bound from O(n2m¯) to O(nm), where n, m, and m¯ are the number of vertices, edges, and non-edges of the graph, respectively. The algorithm uses the vertex ordering characterization that a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices containing both an alternating orientation of the graph and a transitive orientation of the complement of the graph. We also show, as a byproduct, that an alternating orientation can be obtained in O(nm) time for cocomparability graphs, and it is NP-complete to decide whether a graph has an orientation that is alternating and acyclic.

Minimizing crossings in constrained two-sided circular graph layouts.

Circular graph layout is a popular drawing style, in which vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts, which draw some edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all vertices placed on a line (the spine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal $k$-plane set of exteriorly drawn edges for $k \ge 1$, extending the previously studied case $k=0$. We show that this relates to finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary $k$, present an efficient algorithm for $k=1$, and generalize it to an explicit XP-time algorithm for any fixed $k$. For the practically interesting case $k=1$ we implemented our algorithm and present experimental results that confirm its applicability.

On the automatic and a priori design of unstructured mesh resolution for coastal ocean circulation models

This study investigates the design of unstructured mesh resolution and its impact on the modeling of barotropic tides along the United States East Coast and Gulf Coast (ECGC). A discrete representation of a computational ocean domain (mesh design) is necessary due to finite computational resources and an incomplete knowledge of the physical system (e.g., shoreline and seabed topography). The selection of mesh resolution impacts both the numerical truncation error and the approximation of the system’s physical domain. To increase confidence in the design of high-resolution coastal ocean meshes and to quantify the efficacy of current mesh design practices, an automated mesh generation approach is applied to objectively control resolution placement based on a priori information such as shoreline geometry and seabed topographic features. The simulated harmonic tidal elevations for each mesh design are compared to that of a reference solution, computed on a 10.8 million vertex mesh of the ECGC region with a minimum shoreline resolution of 50 m. Our key findings indicate that existing mesh designs that use uniform resolution along the shoreline and slowly varying resolution sizes on the continental shelf inefficiently discretize the computational domain. Instead, a targeted approach that places fine resolution in narrow geometric features, along steep topographic gradients, and along pronounced submerged estuarine channels, while aggressively relaxing resolution elsewhere, leads to a mesh with an order of magnitude fewer vertices than the reference solution with comparable accuracy (within 3% harmonic elevation amplitudes in 99% of the domain).

Timing-Anomaly Free Dynamic Scheduling of Conditional DAG Tasks on Multi-Core Systems

In this paper, we propose a novel approach to schedule conditional DAG parallel tasks, with which we can derive safe response time upper bounds significantly better than the state-of-the-art counterparts. The main idea is to eliminate the notorious timing anomaly in scheduling parallel tasks by enforcing certain order constraints among the vertices, and thus the response time bound can be accurately predicted off-line by somehow “simulating” the runtime scheduling. A key challenge to apply the timing-anomaly free scheduling approach to conditional DAG parallel tasks is that at runtime it may generate exponentially many instances from a conditional DAG structure. To deal with this problem, we develop effective abstractions, based on which a safe response time upper bound is computed in polynomial time. We also develop algorithms to explore the vertex orders to shorten the response time bound. The effectiveness of the proposed approach is evaluated by experiments with randomly generated DAG tasks with different parameter configurations.