Star Subgraphs Research Articles

Complexity of the emptiness problem for graph-walking automata and for tilings with star subgraphs

This paper proves the decidability of the emptiness problem for two models which recognize finite graphs: graph-walking automata, and tilings of graphs by star subgraphs (star automata). Furthermore, it is proved that the non-emptiness problem for graph-walking automata (that is, whether a given automaton accepts at least one graph) is NEXP-complete. For star automata, which generalize nondeterministic tree automata to the case of graphs, it is proved that their non-emptiness problem is NP-complete.

RDF Subgraph Matching by Means of Star Decomposition

<p>With the continuous development of the network, the scale of RDF data is becoming larger and larger. In the face of large-scale RDF data processing, the traditional database query method has been unable to meet the needs. Due to the limited characteristics of subgraph matching, most existing algorithms often have the phenomenon that many subgraphs are repeatedly traversed during the query process, resulting in a large number of intermediate result sets and low query efficiency. The core problem to be solved is how to efficiently match subgraphs. In order to improve the query efficiency of RDF subgraphs in massive RDF data graphs and solve the problem of repeated calculation of some graphs in the query process of RDF subgraphs, an RDF subgraph query algorithm based on star decomposition is proposed in this paper. The algorithm uses graph structure to decompose RDF subgraphs into stars and uses a custom node cost model to calculate the query order of the star subgraphs. By decomposing, the amount of communication among subgraphs is reduced, and the communication cost for query processing is lowered. Moreover, utilizing the query order for RDF subgraph matching can effectively reduce the generation of intermediate result sets and accelerate the efficiency of subgraph matching. On this basis, the performances of the proposed algorithm and several other widely used algorithms are compared and analyzed on two different datasets. Experiments show that the proposed algorithm has better advantages in database recreation, memory size, and execution efficiency. </p> <p> </p>

Total tessellation cover: Bounds, hardness, and applications

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V ( G ) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E ( G ) . A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t ( G ) is the size of a minimum total tessellation cover of G . We present lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ s ( G ) + 1 , where ω ( G ) is the size of a maximum clique, and s ( G ) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t ( G ) = ω ( G ) or T t ( G ) = s ( G ) + 1 . We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t ( G ) ≤ k . We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, ( 2 , 1 ) -chordal graphs and planar graphs.

An Efficient Algorithm of Star Subgraph Queries on Urban Traffic Knowledge Graph

Knowledge graph has wide applications in the field of computer science. In the knowledge service environment, the information is large and explosive, and it is difficult to find knowledge of common phenomena. The urban traffic knowledge graph is a knowledge system that formally describes urban traffic concepts, entities and their interrelationships. It has great application potential in application scenarios such as user travel, route planning, and urban planning. This paper first defines the urban traffic knowledge graph and the star subgraph query of the urban traffic knowledge graph. Then, the road network data and trajectory data are collected to extract the urban traffic knowledge, and the urban traffic knowledge graph is constructed with this knowledge. Finally, a star subgraph query algorithm on the urban traffic knowledge graph is proposed. The discussion of the star subgraph query mode gives the corresponding application scenarios of our method in the urban traffic knowledge graph. Experimental results verify the performance advantages of this method.

The Power Graph of a Dihedral Group

Graph theory is one of the topics in mathematics that is quite interesting to study because it is applicable and can be combined with other mathematical topics such as group theory. The combination of graph theory and group theory is that graphs can be used to represent a group. An example of a graph is a power graph. A power graph of the group is defined as a graph whose vertex set is all elements of and two distinct vertices and are connected if and only if or for a positive integer x and y. In this study, the author discusses the power graph of the dihedral group The results obtained from this study are the power graph of the dihedral group where with prime numbers and an natural number is a graph consisting of two non-disjoint subgraphs, namely complete subgraphs and star subgraphs. And we find that its radius and diameter are 1 and 2.

Susceptible-infected-susceptible model on networks with eigenvector localization.

It is a longstanding debate on the absence of threshold for susceptible-infected-susceptible (SIS) model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network gives rise to the inactive Griffiths phase featuring slow decay of the activity localized around highly connected nodes due to the dynamical fluctuation. We show how it dramatically changes our understanding of the SIS model, opening up new possibilities for the debate. We derive the critical condition for Griffiths to active phase transition: on average, an infected node can further infect another one in the characteristic lifespan of the star subgraph composed of the node and its nearest neighbors. The system approaches the critical point of avoiding the irreversible dynamical fluctuation and the trap of absorbing state. As a signature of the phase transition, the infection density of a node is not only proportional to its degree, but also proportional to the exponentially growing lifespan of the star. And the divergence of the average lifespan of the stars is responsible for the vanishing threshold in the thermodynamic limit. The eigenvector localization exponentially reinforces the infection of highly connected nodes, while it inversely suppresses the infection of small-degree nodes.

On the multiplicity of Laplacian eigenvalues and Fiedler partitions

In this paper we investigate the relation between eigenvalue distribution and graph structure of two classes of graphs: the (m,k)-stars and l-dependent graphs. We give conditions on the topology and edge weights in order to get values and multiplicities of Laplacian matrix eigenvalues. We prove that a vertex set reduction on graphs with (m,k)-star subgraphs is feasible, keeping the same eigenvalues with reduced multiplicity. Moreover, some useful eigenvectors properties are derived up to a product with a suitable matrix. Finally, we relate these results with Fiedler spectral partitioning of the graph and the physical relevance of the results is shortly discussed.

Two-class support vector machine with new kernel function based on paths of features for predicting chemical activity

Information and computer science fields such as machine learning and graph theory are implemented in chemoinformatics to discover the properties of chemical compounds. This paper presents a new algorithm based on the two-class support vector machine (SVM) model, which has new kernel functions for paths of features, enabling the prediction of chemical compound activity. Initially, we extract all paths of features (star subgraphs) with certain lengths, and we encode them depending on their structure in the graphs. Then, we use these codes to construct two relationship matrices between those paths. These matrices contain common and different sub-paths between paths of stars. The number of sub-paths/paths for each compound is passed to the proposed kernel functions in the two-class SVM to predict the activity of chemical compounds. The relationship matrices created by the proposed algorithm help to reduce the number of features, which improves prediction accuracy. We apply the proposed algorithm with and without feature selection using two benchmark datasets, specifically, the monoamine oxidase (MAO) dataset and the AIDS antiviral screen database of active compound dataset, which have 68 and 2000 chemical compounds, respectively. We perform comparative experiments for the proposed kernel functions and many other two-class SVM prediction methods, and the results before feature selection show prediction accuracies of 94% and 99.5% for MAO and AIDS, respectively. After selection, the prediction accuracies are 96% and 99.5% for MAO and AIDS, respectively.

Sublinear-Time Algorithms for Counting Star Subgraphs via Edge Sampling

We study the problem of estimating the value of sums of the form $$S_p \triangleq \sum \left( {\begin{array}{c}x_i p\end{array}}\right) $$Spźźxip when one has the ability to sample $$x_i \ge 0$$xiź0 with probability proportional to its magnitude. When $$p=2$$p=2, this problem is equivalent to estimating the selectivity of a self-join query in database systems when one can sample rows randomly. We also study the special case when $$\{x_i\}$${xi} is the degree sequence of a graph, which corresponds to counting the number of p-stars in a graph when one has the ability to sample edges randomly. Our algorithm for a $$(1 \pm \varepsilon )$$(1?ź)-multiplicative approximation of $$S_p$$Sp has query and time complexities $$\mathrm{O}\left( \frac{m \log \log n}{\epsilon ^2 S_p^{1/p}}\right) $$Omloglogn∈2Sp1/p. Here, $$m=\sum x_i/2$$m=źxi/2 is the number of edges in the graph, or equivalently, half the number of records in the database table. Similarly, n is the number of vertices in the graph and the number of unique values in the database table. We also provide tight lower bounds (up to polylogarithmic factors) in almost all cases, even when $$\{x_i\}$${xi} is a degree sequence and one is allowed to use the structure of the graph to try to get a better estimate. We are not aware of any prior lower bounds on the problem of join selectivity estimation. For the graph problem, prior work which assumed the ability to sample only vertices uniformly gave algorithms with matching lower bounds (Gonen et al. in SIAM J Comput 25:1365---1411, 2011). With the ability to sample edges randomly, we show that one can achieve faster algorithms for approximating the number of star subgraphs, bypassing the lower bounds in this prior work. For example, in the regime where $$S_p\le n$$Sp≤n, and $$p=2$$p=2, our upper bound is $$\tilde{O}(n/S_p^{1/2})$$O~(n/Sp1/2), in contrast to their $$\varOmega (n/S_p^{1/3})$$Ω(n/Sp1/3) lower bound when no random edge queries are available. In addition, we consider the problem of counting the number of directed paths of length two when the graph is directed. This problem is equivalent to estimating the selectivity of a join query between two distinct tables. We prove that the general version of this problem cannot be solved in sublinear time. However, when the ratio between in-degree and out-degree is bounded--or equivalently, when the ratio between the number of occurrences of values in the two columns being joined is bounded--we give a sublinear time algorithm via a reduction to the undirected case.

The Game-Theoretic Formation of Interconnections Between Networks

We introduce a network design game where the objective of the players is to design the interconnections between the nodes of two different networks $G_{1}$ and $G_{2}$ in order to maximize certain local utility functions. In this setting, each player is associated with a node in $G_{1}$ and has functional dependencies on certain nodes in $G_{2}$ . We use a distance-based utility for the players in which the goal of each player is to purchase a set of edges (incident to its associated node) such that the sum of the distances between its associated node and the nodes it depends on $G_{2}$ is minimized. We consider a heterogeneous set of players (i.e., players have their own costs and benefits for constructing edges). We show that finding a best response of a player in this game is NP-hard. Despite this, we characterize some properties of the best response actions, which are helpful in determining a Nash equilibrium for certain instances of this game. In particular, we prove the existence of pure Nash equilibria in this game when $G_{2}$ contains a star subgraph, and provide an algorithm that outputs such an equilibrium for any set of players. Finally, we show that the price of anarchy in this game can be arbitrarily large.

Graph Characterization by Counting Sink Star Subgraphs

In this paper, we study the polynomial coefficients of the reduced Bartholdi zeta function for characterizing simple unweighted graphs and demonstrate how to use these coefficients for clustering graphs. The polynomial coefficients of the reduced Bartholdi zeta function are invariant to vertex order permutations and also carry information about counting the sink star subgraphs in a symmetric digraph of G. We also investigate the advantages of the reduced Bartholdi coefficients over other spectral methods such as the Ihara zeta function and Laplacian spectra. Experimental results indicate that the proposed method is more effective than the other spectral approaches, and compared to the Ihara zeta function, it has less sensitivity to structural noises such as omitting an edge.

The coefficients of the reduced Bartholdi zeta function

In this paper, we establish a new zeta function based on the Bartholdi zeta function for an undirected graph G called the reduced Bartholdi zeta function. We study the relation between its coefficients and the structure of the graph, and demonstrate that the coefficients count the star subgraphs in the symmetric digraph D(G). Moreover, we investigate the properties of semi-principle minors extracted from the adjacency matrix of the oriented line graph of G. We also present a general formula for calculating all the coefficients of the reduced Bartholdi zeta function.

Disjoint Hamilton cycles in transposition graphs

Most network topologies that have been studied have been subgraphs of transposition graphs. Edge-disjoint Hamilton cycles are important in network topologies for improving fault-tolerance and distribution of messaging traffic over the network. Not much was known about edge-disjoint Hamilton cycles in general transposition graphs until recently Hung produced a construction of 4 edge-disjoint Hamilton cycles in the 5-dimensional transposition graph and showed how edge-disjoint Hamilton cycles in (n+1)-dimensional transposition graphs can be constructed inductively from edge-disjoint Hamilton cycles in n-dimensional transposition graphs. In the same work it was conjectured that n-dimensional transposition graphs have n−1 edge-disjoint Hamilton cycles for all n greater than or equal to 5. In this paper we provide an edge-labelling for transposition graphs and, by considering known Hamilton cycles in labelled star subgraphs of transposition graphs, are able to provide an extra edge-disjoint Hamilton cycle at the inductive step from dimension n to n+1, and thereby prove the conjecture.

Results on the regularity of square-free monomial ideals

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E∖F|≤1. The Castelnuovo–Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

The performance of an upper bound on the fractional chromatic number of weighted graphs

Given a weighted graph G x , where ( x ( v ) : v ∈ V ) is a non-negative, real-valued weight assigned to the vertices of G , let B ( G x ) be an upper bound on the fractional chromatic number of the weighted graph G x ; so χ f ( G x ) ≤ B ( G x ) . We consider a particular upper bound B resulting from a generalization of the greedy coloring algorithm to weighted graphs. To investigate the worst-case performance of this upper bound, we study the graph invariant β ( G ) = sup x ≠ 0 B ( G x ) χ f ( G x ) . This graph invariant is shown to be equal to the size of the largest star subgraph in the graph. These results have implications for the design and performance of distributed communication networks.

An Airspace-Planning and Collaborative Decision-Making Model: Part II—Cost Model, Data Considerations, and Computations

In Part I of this paper, we presented a large-scale airspace-planning and collaborative decision-making (APCDM) model that is part of a Federal Aviation Administration (FAA)-sponsored effort to enhance the management of the National Airspace System (NAS). Given a set of flights that must be scheduled during some planning horizon, along with alternative surrogate trajectories for each flight, we developed a mixed-integer programming model to select a set of flight plans from among these alternatives, subject to flight safety, air-traffic control workload, and airline equity considerations. The present paper offers insights related to, and a detailed description of, implementing this APCDM model, including the development of a comprehensive cost model, a study for prescribing a set of appropriate parameter values for the overall model, and an investigation on incorporating a suitable set of valid inequalities in the model formulation. Computational results are presented based on several test cases derived from the Enhanced Traffic Management System (ETMS) data provided by the FAA. The results indicate that under plausible probabilistic trajectory error assumptions and with the incorporation of star subgraph convex hull-based valid inequalities, the model offers a viable tool that can be used by the FAA for both tactical and strategic applications.

Bell-type inequalities embedded in the subgraph of graph states

We investigate the Bell-type inequalities of graph states. In this paper, Bell-type inequalities can be derived based on two kinds of the associated subgraphs of the graph states. First, the star subgraphs lead to the maximal violation of the modified Seevinck-Svetlichny inequalities. Second, cycle subgraphs lead to maximal violation of Bell-type inequalities. As a result, once the associated graph of a graph state is given, the corresponding Bell operators can be immediatedly determined using stabilizing generators. In the above Bell-type inequalities, two measurement settings for each party are required.

Obtaining online approximation algorithms for facility dispersion from offline algorithms

AbstractFacility dispersion problems arise in the context of placing obnoxious facilities and retail outlets. In the offline version of such a problem, the input consists of a complete graph on n nodes, a nonnegative weight (distance) for each edge, and the number k ⩽ n of facilities to be placed. The goal is to choose a facility placement consisting of k nodes so as to maximize a given measure of the distances among the facilities. Here, we consider an online version of the problem where the value of k is not known apriori; instead, requests for facilities arrive one at a time. It is also required that previously placed facilities cannot be moved or eliminated. Our main result is that for any objective that satisfies two properties, namely monotonicity and graceful degradation, any offline approximation algorithm with a performance guarantee ρ can be used to develop an algorithm with competitive ratio cρ for the online version, where c is a constant independent of the problem instance. Objectives for which our result applies include the average edge weight and average weight of a star subgraph. The result holds even when the edge weights do not satisfy the triangle inequality. We also identify dispersion objectives for which the offline and online versions have different behaviors when only one of the above two properties is satisfied. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(4), 206–217 2006

A star-based model for the eigenvalue power law of Internet graphs

Using a simple deterministic model for the Internet graph we show that the eigenvalue power-law distribution for its adjacency matrix is a direct consequence of the degree distribution and that the graph must contain many star subgraphs.

Optimal factorizations of families of trees

Let { T 1,…, T k } be a set of trees which is K h -packable. It is shown that every n-vertex graph G=( V, E) with δ(G)⩾n/2+3h n log n has k subgraphs S 1,…, S k with the following properties: 1. S i is a set of ⌊ n/ h⌋ vertex-disjoint copies of T i . 2. The subgraphs S 1,…, S k are edge-disjoint. 3. S 1∪⋯∪ S k has maximum degree at most h−1. There are many interesting special cases of this result. To name just two: • If H is a tree with h vertices and G=( V, E) is a graph with n vertices, h divides n, and δ(G)⩾n/2+3h n log n , then G has an H-factor. • If h divides n, and δ(G)⩾n/2+3h n log n , then G has a set S of n star subgraphs, where for each i=1,…, h, there are exactly n/ h stars in S having i vertices, any two members of S having the same size are vertex-disjoint, and the union of all the members of S is an h−1 regular spanning subgraph of G.