Minimal Graphs Research Articles

MAGD: Minimal Attack Graph Generation Dynamically in Cyber Security

Cyber security faces challenges in detecting and mitigating complex attacks. Security solutions have employed Attack Graphs (AGs) for modeling multi-stage attacks, but traditional AGs suffer from scalability issues and may miss new vulnerabilities and attack paths. Also, traditional AGs construct the graph using information about previously known attacks. In this paper, we propose Minimal Attack Graph Generation Dynamically (MAGD), which leverages data from a deception system based on Honeypots to generate a minimal AG dynamically. In this paper, the AG has been constructed from real-time attacker’s behavior data directly. In addition, MAGD specifically focuses on modeling the attacker’s behavior at the host level, in contrast to traditional network-based AGs that encompass all possible attack paths at the network level. MAGD contains three custom algorithms to construct attacker behavior, generate a minimal AG, and continuously update the graph with new attack information. Complexity analyses demonstrate that MAGD’s generation process can accomplish within polynomial time. Our approach offers several advantages over traditional AGs, including the ability to model attackers’ real-time behavior, construct attackers’ action paths in the target host, and detect new vulnerabilities and attack paths in the victim host. Also, MAGD includes information about the effects of the actions in the target system. This information can be used for other security purposes. We demonstrate MAGD’s efficacy through a case study. MAGD provides a more effective way to detect and mitigate cyber threats by utilizing Honeypot data and proposed algorithms.

Reconstruction from one labelled card and more

The deck D of a graph H is its multiset of one-vertex deleted subgraphs. Ulam's Reconstruction Conjecture claims that every graph having more than two vertices is reconstructible from D. We seek minimal graph parameters that need to accompany one card G to enable the unique reconstruction of H. A card needs to be feasible. We consider specific graph invariants associated with the 0–1 eigenspaces of H, which need to supplement a feasible card, for the unique reconstruction of H. Non–isomorphic graphs sharing a common card turn out to have linearly independent one-dimensional eigenspaces, whereas non–isomorphic graphs, sharing a common eigenspace, have no labelled card in common. Moreover, we determine certain families of graphs for which reconstruction is possible from any card.

Forbidden pairs of disconnected graphs for supereulerianity of connected graphs

Let H be a set of graphs. A graph G is said to be H-free if G does not contain H as an induced subgraph for all H∈H, and we call H a forbidden pair if |H|=2. A graph is called supereulerian if it contains a spanning connected even subgraph. In 1979, Pulleyblank showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. In this paper, we characterize all pairs of graphs R,S (not necessary connected) such that every 2-connected or 2-edge-connected {R,S}-free graph (of sufficiently large order) is supereulerian. We also characterize all minimal 2-connected non-supereulerian graphs, which extends a result by the third author and Xiong (2017) [29].

A note on Reed's conjecture for triangle-free graphs

Reed's Conjecture states that χ(G)≤⌈(Δ(G)+ω(G)+1)/2⌉, where χ(G), Δ(G) and ω(G) are the chromatic number, maximum degree and clique number of a graph G, respectively.In this note, we prove this conjecture for maximal triangle-free graphs with maximum degree less than 7. Moreover, we show that Reed's Conjecture holds for all graphs with girth at least 5 up to at least 30 vertices and for all triangle-free graphs G up to at least 32 vertices such that χ(G)≠5 which improves similar results given in [Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, J. Graph Theory, 93(2020), 34–48.]

Boundary regularity of minimal graphs in the hyperbolic space

Abstract F.-H. Lin (1989) studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd-dimensional case and study how the higher regularity is obstructed. We introduce the logarithmic distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.

A degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods

We investigate a degree preserving variant of the \Delta -Y transformation which replaces a triangle with a new 6-valent vertex which has double edges to the vertices that had been in the triangle. This operation is relevant for understanding scalar Feynman integrals in 6 dimensions. We study the structure of equivalence classes under this operation and its inverse, with particular attention to when the equivalence classes are finite, when they contain simple 6-regular graphs, and when they contain doubled 3-regular graphs. The last of these, in particular, is relevant for the Feynman integral calculations, and we make some observations linking the structure of these classes to the Feynman periods. Furthermore, we investigate properties of minimal graphs in these equivalence classes.

Gaussian curvature of minimal graphs in [formula omitted

In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold M×R. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via ℘-harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.

Thermal properties of structurally balanced systems on classical random graphs

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) under the influence of the temperature modeling the social noise level are discussed for interacting actors occupying nodes of classical random graphs. Depending on the graph density D, either a smooth crossover or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase in the thermal noise level. The minimal graph density Dmin for which the first-order phase transition can be observed decreases with the system size N as Dmin∝N−0.58(1). For graph densities D>Dmin, the reduced critical temperature Tc⋆=Tc/Tc(D=1) increases with the graph density as Tc⋆∝D1.719(6) independently of the system size N.

Minimal graphs for contractible and dismantlable properties

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be I-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of I-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of k-dismantlable graphs and k-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.

A New Algorithm of Cubic Dynamic Uncertain Causality Graph for Speeding Up Temporal Causality Inference in Fault Diagnosis

Representing and reasoning uncertain causalities have diverse applications in fault diagnosis for industrial systems. Owing to the complicated dynamics and a multitude of uncertain factors in such systems, it is hard to implement efficient diagnostic inference when a process fault occurs. The cubic dynamic uncertain causal graph was proposed for graphically modeling and reasoning about the fault spreading behaviors in the form of causal dependencies across multivariate time series. However, in certain large-scale scenarios with multiconnected and time-varying causalities, the existing inference algorithm is incapable of dealing with the logical reasoning process in an efficient manner. We, therefore, explore the solutions to enhanced computational efficiency. Causality graph decomposition and simplification, and graphical transformation, are proposed to reduce model complexity and form a minimal causality graph. An algorithm, event-oriented early logical absorption, is also presented for logical reasoning. It is mainly intended to minimize the computational costs for compulsory absorption operations in the early stage of reasoning process. The effectiveness of the proposed algorithm is verified on the secondary loop model of nuclear power plant by utilizing the fault data derived from a nuclear power plant simulator. The results show the anticipated capability of efficient fault diagnosis of the proposed algorithm for large-scale dynamic systems.

On Graphs with c2-c3 Successive Minimal Laplacian Coefficients

Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as P(G;λ)=det(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is called the i-th Laplacian coefficient of G. Denoted by Gn,m the set of all (n,m)-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each i(i=0,1,…,n), H is ci(G)-minimal in Gn,m. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in Gn,m because such graphs have minimal Laplacian-like energy. We determine that the c2(G)-c3(G) successive minimal graph is exactly one of the four classes of threshold graphs.

Quantum routing of information using chiral quantum walks

We address routing of classical and quantum information over quantum network and show how to exploit chirality (directionality) to achieve nearly optimal and robust transport. In particular, we prove how continuous-time chiral quantum walks over a minimal graph are able to model directional transfer of information over a network. At first, we show how classical information, encoded onto an excitation localized at one vertex of a simple graph, may be sent to any other chosen location with nearly unit fidelity by tuning a single phase. Then, we prove that high-fidelity transport is also possible for coherent superpositions of states, i.e., for routing of quantum information. Furthermore, we show that by tuning the phase parameter, one obtains universal quantum routing, i.e., independent on the input state. In our scheme, chirality is governed by a single phase, and the routing probability is robust against fluctuations of this parameter. Finally, we address characterization of quantum routers and show how to exploit the self-energies of the graph to achieve high precision in estimating the phase parameter.

On Ramsey (C4, K1, n)-minimal graphs

Let F , G and H be any simple graphs. The notation F → ( G , H ) means for any red-blue coloring on the edges of graph F , there exists either a red copy of G or a blue copy of H . If F → ( G , H ) , then graph F is called a Ramsey graph for ( G , H ) . Additionally, if the graph F satisfies that F − e ↛ ( G , H ) for any edge e of F , then graph F is called a Ramsey ( G , H ) -minimal. The set of all Ramsey ( G , H ) -minimal graphs is denoted by ℛ( G , H ) . In this paper, we construct a new class of Ramsey ( C 4 , K 1, n ) -minimal graphs.

An analogue of Aldous’s theorem on mixing times of a random walk for complex reflection groups

The subject of this paper is the mixing time of random walks on minimal Cayley graphs of complex reflection groups G(m,1,n). The key role in estimating it is played by the coupling of distributions, which has been used before for the same task on symmetric groups. The difficulty with its adaptation for the current case is that there are now two components in a walk, which are to be coupled, and they influence each other’s behaviour. To solve this problem, random walks are split into several blocks for each of which the time needed for their states to match is estimated separately. The result is upper and lower bounds on mixing times of random walks on complex reflection groups, analogous to those obtained by Aldous for a symmetric group.

Minimal bipartite dimers and higher genus Harnack curves

This paper completes the comprehensive study of the dimer model on infinite minimal graphs with Fock's weights [arXiv:1503.00289] initiated in [arXiv:2007.14699]: the latter article dealt with the elliptic case, i.e., models whose associated spectral curve is of genus one, while the present work applies to models of arbitrary genus. This provides a far-reaching extension of the genus zero results of [arXiv:math-ph/0202018, arXiv:math/0311062], from isoradial graphs with critical weights to minimal graphs with weights defining an arbitrary spectral data. For any minimal graph with Fock's weights, we give an explicit local expression for a two-parameter family of inverses of the associated Kasteleyn operator. In the periodic case, this allows us to prove local formulas for all ergodic Gibbs measures, thus providing an alternative description of the measures constructed in [arXiv:math-ph/0311005]. We also compute the corresponding slopes, exhibit an explicit parametrization of the spectral curve, identify the divisor of a vertex, and build on [arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between Fock's models on periodic minimal graphs and Harnack curves endowed with a standard divisor.

Elliptic Dimers on Minimal Graphs and Genus 1 Harnack Curves

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock’s elliptic weights (Fock, Inverse spectral problem for GK integrable system. arXiv e-prints arXiv:1503.00289 , 2015). Specific instances of such models were studied in Boutillier et al. (Invent Math 208(1):109–189, 2017), Boutillier et al. (Probability theory and related fields, 2018) and de Tilière (Electron J Probab 26:1–86, 2021); we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of Kenyon (Invent Math 150(2):409–439, 2002) and Kenyon and Okounkov (Duke Math J 131(3):499–524, 2006) on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $$\mathbb {Z}^2$$ -periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in Kenyon et al. (Ann Math 163(3):1019–1056, 2006). We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay’s trisecant identity.

On the Minimum Degree of Minimal Ramsey Graphs for Cliques Versus Cycles

A graph is said to be -Ramsey for a -tuple of graphs , denoted by , if every -edge-coloring of contains a monochromatic copy of in color for some . Let denote the smallest minimum degree of over all graphs that are minimal -Ramsey for (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, Erdős, and Lovász, who determined its value precisely for a pair of cliques. Over the past two decades the parameter has been studied by several groups of authors, their main focus being on the symmetric case, where for all . The asymmetric case, in contrast, has received much less attention. In this paper, we make progress in this direction, studying asymmetric tuples consisting of cliques, cycles, and trees. We determine when is a pair of one clique and one tree, a pair of one clique and one cycle, and a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on in terms of the size of the cliques , the number of cycles, and the number of cliques. Our bounds are tight up to logarithmic factors when two of the three parameters are fixed.

On Minimum Generalized Degree Distance Index of Cyclic Graphs

Topological index (TI) is a mapping that associates a real number to the under study (molecular) graph which predicts its various physical and chemical properties. The generalized degree distance index is the latest developed TI having compatible significance among the list of distance‐based TIs. In this paper, the minimum generalized degree distance of unicyclic, bicyclic, and four cyclic graphs is determined. Mainly, the associated extremal (minimal) graphs are also identified among all the aforesaid classes of graphs.

The stickiness property for antisymmetric nonlocal minimal graphs

<p style='text-indent:20px;'>We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants).</p><p style='text-indent:20px;'>In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.</p>

Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian {K1,3,N3,1,1}-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs.