Graph Realization Research Articles

Graph realization of distance sets

The Distance Realization problem is defined as follows. Given an n×n matrix D of nonnegative integers, interpreted as inter-vertex distances, find an n-vertex weighted or unweighted graph G realizing D, i.e., whose inter-vertex distances satisfy distG(i,j)=Di,j for every 1≤i<j≤n, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of Distance Realization that was studied in the past is where each entry in the matrix D may contain a range of consecutive permissible values. We refer to this extension as Range Distance Realization (or Range-DR). Restricting each range to at most k values yields the problem k-Range Distance Realization (or k-Range-DR). The current paper introduces a new extension of Distance Realization, in which each entry Di,j of the matrix may contain an arbitrary set of acceptable values for the distance between i and j, for every 1≤i<j≤n. We refer to this extension as Set Distance Realization (Set-DR), and to the restricted problem where each entry may contain at most k values as k-Set Distance Realization (or k-Set-DR).We first show that 2-Range-DR is NP-hard for unweighted graphs (implying the same for 2-Set-DR). Next we prove that 2-Set-DR is NP-hard for unweighted and weighted trees.Finally, we explore Set-DR where the realization is restricted to the families of stars, paths, cycles, or caterpillars. For the weighted case, our positive results are that there exist polynomial time algorithms for the 2-Set-DR problem on stars, paths and cycles, and for the 1-Set-DR problem on caterpillars. On the hardness side, we prove that 6-Set-DR is NP-hard for stars and 5-Set-DR is NP-hard for paths, cycles and caterpillars. For the unweighted case, our results are the same, except for the case of unweighted stars, for which k-Set-DR is polynomially solvable for any k.

Graph realization of sets of integers

Graph theory is used in many areas of chemical sciences, especially in molecular chemistry. It is particularly useful in the structural analysis of chemical compounds and in modeling chemical reactions. One of its applications concerns determining the structural formula of a chemical compound. This can be modeled as a variant of the well-known graph realization problem. In the classical version of the problem, a sequence of natural numbers is given, and the question is whether there exists a graph in which the vertices have degrees equal to the given numbers. In the variant considered in this paper, instead of a sequence of natural numbers, a sequence of sets of natural numbers is given, and the question is whether there exists a multigraph such that each of its vertices has a degree equal to a number from one of the sets. This variant of the graph realization problem matches the nature of the problem of determining the structural formula of a chemical compound better than other variants considered in the literature. We propose a polynomial time exact algorithm solving this variant of the problem.

Bond graph realizations of state space linear time-invariant multivariable descriptions

Given a state space description of a linear time-invariant multivariable system, a similarity transformation is applied to get a controllability canonical form. This transformation assures a Bond Graph (BG) realization and avoids the use of active bonds. Active or signal bonds do not preserve the continuity of power flows and consequently, energy conservation properties are lost. The state matrix of the system in transformed coordinates contains an antisymmetric matrix associated with the storage field and the rest is associated with the dissipative field. Thus, in general, the dissipative field is implemented with a multiport element, and in order to give physical meaning, the storage field is implemented with one-port elements. The resulting BG model does not have zero-order causal paths, that is, algebraic loops between the dissipative elements. The results are applied to the BG realization of a linear robust multivariable controller that is designed for a two mass system modelled in BG.

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induce a P S L ( 2 , C ) PSL(2,\mathbb {C}) -valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same discrete conformal structure. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H ≡ 1 H\equiv 1 in hyperbolic space. We further establish convergence on triangular lattices.

A mathematical model of thermoplastic elastomers for analysing the topology of microstructures and mechanical properties during elongation

In this study, a mathematical model based on graph theory is developed to analyse the deformed structures and mechanical properties of thermoplastic elastomers (TPEs) using ABA-type triblock copolymers. TPEs exhibit a network structure formed by bridge chains; deformation of this network structure causes stress. During the deformation of TPEs, domain breakage and coalescence occur, accompanied by topological changes in the chains, such as conformational transitions between the bridge and loop chains. By employing the mathematical concepts of harmonic realization of graphs in the physical space and the tension tensor to quantify the stress in the bridge-chain network structure, an effective method for analysing topologicalchanges in microstructures caused by elongation is proposed. As an application of this method, optimal geometric structures of block copolymers with desired functionalities can be determined.

Approximate Sampling of Graphs with Near-P-Stable Degree Intervals

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (DOI:10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences.

Methods for the graph realization problem

Abstract The graph realization problem seeks an answer to how and under what conditions a graph can be constructed if we know the degrees of its vertices. The problem was widely studied by many authors and in many ways, but there are still new ideas and solutions. In this sense, the paper presents the known necessary and su cient conditions for realization with the description in pseudocode of the corresponding algorithms. Two cases to solve the realization problem are treated: finding one solution, and finding all solutions. In this latter case a parallel approach is presented too, and how to exclude isomorphic graphs from solutions. We are also discussing algorithms using binary integer programming and flow networks. In the case of a bigraphical list with equal out- and in-degree sequences a modified Edmonds–Karp algorithm is presented such that the resulting graph will be always symmetric without containing loops. This algorithm solves the problem of graph realization in the case of undirected graphs using flow networks.

A Multi-Objective Degree-Based Network Anonymization Method

Enormous amounts of data collected from social networks or other online platforms are being published for the sake of statistics, marketing, and research, among other objectives. The consequent privacy and data security concerns have motivated the work on degree-based data anonymization. In this paper, we propose and study a new multi-objective anonymization approach that generalizes the known degree anonymization problem and attempts at improving it as a more realistic model for data security/privacy. Our suggested model guarantees a convenient privacy level, based on modifying the degrees in a way that respects some given local restrictions, per node, such that the total modifications at the global level (in the whole graph/network) are bounded by some given value. The corresponding multi-objective graph realization approach is formulated and solved using Integer Linear Programming to obtain an optimum solution. Our thorough experimental studies provide empirical evidence of the effectiveness of the new approach, by specifically showing that the introduced anonymization algorithm has a negligible effect on the way nodes are clustered, thereby preserving valuable network information while significantly improving data privacy.

Coupler curves of moving graphs and counting realizations of rigid graphs

A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.

Recovering the Graph Underlying Networked Dynamical Systems under Partial Observability: A Deep Learning Approach

We study the problem of graph structure identification, i.e., of recovering the graph of dependencies among time series. We model these time series data as components of the state of linear stochastic networked dynamical systems. We assume partial observability, where the state evolution of only a subset of nodes comprising the network is observed. We propose a new feature-based paradigm: to each pair of nodes, we compute a feature vector from the observed time series. We prove that these features are linearly separable, i.e., there exists a hyperplane that separates the cluster of features associated with connected pairs of nodes from those of disconnected pairs. This renders the features amenable to train a variety of classifiers to perform causal inference. In particular, we use these features to train Convolutional Neural Networks (CNNs). The resulting causal inference mechanism outperforms state-of-the-art counterparts w.r.t. sample-complexity. The trained CNNs generalize well over structurally distinct networks (dense or sparse) and noise-level profiles. Remarkably, they also generalize well to real-world networks while trained over a synthetic network -- namely, a particular realization of a random graph.

Stable Bi-Maps on Surfaces and Their Graphs

In this paper we study stable bi-maps F = (f1, f2): M →R×R^2 from a global viewpoint,where M is a smooth closed orientable surface and f1: M→R, f2: M→R^2 are stable maps.We associate a graph to F, so-called RM-graph and study its properties. The RM-graph captures more information about the topological structure of M than other graphs that appear in literature. Moreover, some graph realization theorems are obtained.

Graph realizations: Maximum degree in vertex neighborhoods

This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

On Reconfiguration Graphs of Independent Sets Under Token Sliding

An independent set of a graph G is a vertex subset I such that there is no edge joining any two vertices in I. Imagine that a token is placed on each vertex of an independent set of G. The $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) reconfiguration graph of G takes all non-empty independent sets (of size k) as its nodes, where k is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph G: (1) Whether the $${\textsf{TS}}_k$$ -reconfiguration graph of G belongs to some graph class $${\mathcal {G}}$$ (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If G satisfies some property $${\mathcal {P}}$$ (including s-partitedness, planarity, Eulerianity, girth, and the clique’s size), whether the corresponding $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) reconfiguration graph of G also satisfies $${\mathcal {P}}$$ , and vice versa. Additionally, we give a decomposition result for splitting a $${\textsf{TS}}_k$$ -reconfiguration graph into smaller pieces.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ are adjacent in $$\Gamma (R)$$ if and only if $$xy=0$$ . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ , $$1 \le i \le k$$ are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ , where $$Aut(\Gamma (R))$$ denotes the automorphism group of $$\Gamma (R)$$ . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ is a connected threshold graph if and only if $$R\cong {F}_{q}$$ or $$R\cong {F}_2 \times {F}_{q}$$ . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

Reeb graphs of circle-valued functions: A survey and basic facts

The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties. However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application. In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces. We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions. In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.

Blume–Emery–Griffiths model on random graphs

The Blume–Emery–Griffiths model with a random crystal field is studied in a random graph architecture, in which the average connectivity is a controllable parameter. The disordered average over the graph realizations is treated by replica symmetry formalism of order parameter functions. A self consistent equation for the distribution of local fields is derived, and numerically solved by a population dynamics algorithm. The results show that the average connectivity amounts to changes in the topology of the phase diagrams. Phase diagrams for representative values of the model parameters are compared with those obtained for fully connected mean field and renormalization group approaches.

Realization of a graph as the Reeb graph of a height function on an embedded surface

We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.

Dynamic Modelling and Numerical Simulation of Formation Control for Intelligent Multi-agent System with Target Geometric Configuration

Swarming motility arise very naturally in biological, physical, social sciences, etc. However, how to realize artificial intelligent self-organizing behavior is still an interesting and challenging task, especially for formation control of multiple agents with special geometric configuration. This work proposes a novel approach of formation control for a multi-agent system with target geometric configuration by combining dynamic model with graph realization. First, the global rigid graph is designed for the target formation pattern, then the interactive relationship and the expected distance between different intelligent agents are identified by the realization of graph. Secondly, the double-integrator dynamic model based on Newtonian mechanics is formulated for inherent driving force caused upon attenuation of potential energy, consistent of movement direction and speed. Thirdly, the stability of swarming motility is proved by Lyapunov’s second method, i.e., the movement of all agents will gradually stabilize to consistency of the movement direction and speed, and realize the target geometric configuration. Finally, numerical simulations of different geometric configurations are performed to verify the theoretical findings.

Counting geodesics of given commutator length

Abstract Let $\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma $ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

Irredundance graphs

A set D of vertices of a graph G = ( V , E ) is irredundant if each v ∈ D satisfies (a) v is isolated in the subgraph induced by D , or (b) v is adjacent to a vertex in V − D that is nonadjacent to all other vertices in D . The upper irredundance number IR ( G ) is the largest cardinality of an irredundant set of G ; an IR ( G ) -set is an irredundant set of cardinality IR ( G ) . The IR -graph of G has the irredundant sets of G of maximum cardinality, that is, the IR ( G ) -sets, as vertex set, and sets D and D ′ are adjacent if and only if D ′ is obtained from D by exchanging a single vertex of D for an adjacent vertex in D ′ . We study the realizability of graphs as IR -graphs and show that all disconnected graphs are IR -graphs, but some connected graphs (e.g. stars K 1 , n , n ≥ 2 , P 4 , P 5 , C 5 , C 6 , C 7 ) are not. We show that the double star S ( 2 , 2 ) – the tree obtained by joining the two central vertices of two disjoint copies of P 3 – is the unique smallest IR -tree with diameter 3 and also a smallest non-complete IR -tree, and the tree obtained by subdividing a single pendant edge of S ( 2 , 2 ) is the unique smallest IR -tree with diameter 4.