Vertex Set Research Articles

Double-Star Isolation of Maximal Outerplanar Graphs ∗

In the graph G = (V, E), V represents the set of vertices and E represents the set of edges. ℱ represents a family of graphs. A subset S ⊆ V is considered an ℱ -isolating set if G[V\\NG[S]] does not contain F as a subgraph for all F ∈ ℱ. The ℱ-isolation number of the graph G, denoted by 𝜄(G, ℱ), is defined as the minimum cardinality of an ℱ-isolating set in G. If the family ℱ consists of a single graph H, then the subset S is referred to as an H-isolating set. The H-isolation number of the graph G, denoted by 𝜄 H (G), is defined as the minimum cardinality of an H-isolating set in G. Maximal outerplanar graphs have been widely applied in different fields of research since 1891, where they hold significant importance. This paper aims to provide a comprehensive analysis of the H-isolation number in maximal outerplanar graphs, , where H ≅ S 1,k+1,k+1.

Positioning errors of anatomical landmarks identified by fixed vertices in homologous meshes

BackgroundHuman movement analysis is usually achieved by tracking markers attached to anatomical landmarks with photogrammetry. Such marker-based systems have disadvantages that have led to the development of markerless procedures, although their accuracy is not usually comparable to that of manual palpation procedures. New motion acquisition systems, such as 3D temporal scanners, provide homologous meshes that can be exploited for this purpose. Research questionCan fixed vertices of a homologous mesh be used to identify anatomical landmarks with an accuracy equivalent to that of manual palpation? MethodsWe used 3165 human shape scans from the CAESAR dataset, with labelled locations of anatomical landmarks. First, we fitted a template mesh to the scans, and assigned a vertex of that mesh to 53 anatomical landmarks in all subjects. Then we defined a nominal vertex for each landmark, as the more centred vertex out of the set assigned for that landmark. We calculated the errors of the template-fitting and the nominal vertex determination procedures, and analysed their relationship to subject’s sex, height and body mass index, as well as their size compared to manual palpation errors. ResultsThe template-fitting errors were below 5 mm, and the nominal vertex determination errors reached maximum values of 24 mm. Except for the trochanter, those errors were the same order of magnitude or smaller than inter-examiner errors of lower limb landmarks. Errors increased with height and body mass index, and were smaller for men than for women of the same height and body mass index. SignificanceWe defined a set of vertices for 53 anatomical landmarks in a homologous mesh, which yields location errors comparable to those obtained by manual palpation for the majority of landmarks. We also quantified how the subject’s sex and anthropometric features can affect the size of those errors.

Densest Multipartite Subgraph Search in Heterogeneous Information Networks

Cohesive multipartite subgraphs (CMS) in heterogeneous information networks (HINs) uncover closely connected vertex groups of multiple types, enhancing real applications like community search and anomaly detection. However, existing works for HINs pay less attention to searching CMS. In this paper, we leverage well-established concepts of meta-path and densest subgraph to propose a novel CMS model called the densest P -partite subgraph. Given a multipartite subgraph of an HIN induced by i =| P | types of vertices defined in a query meta-path P (i.e., a P -partite subgraph), we devise a novel density function which is the number of the instances of P over the geometric mean of the sizes of i different types of vertex sets in the subgraph. A P -partite subgraph with the highest density serves as the optimum result. To find the densest P -partite subgraph in an HIN with n vertices, we first design an exact algorithm with a runtime cost equivalent to solving Θ(|M|) instances of the min-cut problem where |M|= O (( n/i ) i ). Then, we attempt a more efficient approximation algorithm that achieves a ratio of 1/ i but still incurs the cost of solving Θ(|M|) instances of our proposed peeling problem. Both approaches struggle with scalability due to Θ(|M|). To overcome this bottleneck, we improve the exact algorithm with novel pruning rules that non-trivially reduce the number of min-cut problem instances to solve to O (|M|). Empirically, 70-90% instances are pruned, making the improved exact algorithm significantly faster than the approximation algorithm. Extensive experiments on real datasets demonstrate the effectiveness of the proposed model and the efficiency of our algorithms.

Exact algorithms for restricted subset feedback vertex set in chordal and split graphs

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The task is to determine whether there exists a subset S⊆V∖T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP-complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal and split graphs can be solved in O(1.1550|V|) time and exponential space or in O(1.1605|V|) time and polynomial space, significantly improving all previous results. As a by-product, we show that the Maximum Independent Set problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to Vertex Cover, we obtain an O⁎(1.2738k)-time parameterized algorithm and a tight O(k2)-kernel for R-SFVS in chordal and split graphs.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph

The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $\langle u, v\rangle \in P(Z_n)$ are two diverse adjacent vertices if and only if $u \ne v$ and $\langle v \rangle \subseteq \langle u \rangle$ or $\langle u \rangle \subseteq \langle v \rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right), \ \frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right)+\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,{{d}^{2}}\left(u,v \right),$$ $$\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,\frac{1}{d\left(u,v \right)} \mbox{ and } \frac{1}{2}\underset{uv\in E\left( G \right)}{\mathop \sum }\,\left( {{d}_{u}}+{{d}_{v}} \right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and <em>SK</em> the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and <em>SK</em> indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k \cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.

Toward a Better Understanding of Randomized Greedy Matching

There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze [ 9 ]. We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. [ 12 ]. We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al. [ 1 ] that is proved to achieve a (0.5+ε)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm named Random Decision Order (RDO) that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that the RDO algorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio of MRG . Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501-approximation ratio. This result solves the open question by Chan et al. [ 4 ] and Gamlath et al. [ 12 ] about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1-1/ e )-approximation for edge-weighted bipartite graphs, which generalizes the (1-1/ e )-approximation ratio of Gamlath et al. [ 12 ] for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.

Capturing Weak Interactions in Surface Adsorbate Systems at Coupled Cluster Accuracy: A Graph-Theoretic Molecular Fragmentation Approach Improved through Machine Learning.

The accurate and efficient study of the interactions of organic matter with the surface of water is critical to a wide range of applications. For example, environmental studies have found that acidic polyfluorinated alkyl substances, especially perfluorooctanoic acid (PFOA), have spread throughout the environment and bioaccumulate into human populations residing near contaminated watersheds, leading to many systemic maladies. Thus, the study of the interactions of PFOA with water surfaces became important for the mitigation of their activity as pollutants and threats to public health. However, theoretical study of the interactions of such organic adsorbates on the surface of water, and their bulk concerted properties, often necessitates the use of ab initio methods to properly incorporate the long-range electronic properties that govern these extended systems. Notable theoretical treatments of "on-water" reactions thus far have employed hybrid DFT and semilocal DFT, but the interactions involved are weak interactions that may be best described using post-Hartree-Fock theory. Here, we aim to demonstrate the utility of a graph-theoretic approach to molecular fragmentation that accurately captures the critical "weak" interactions while maintaining an efficient ab initio treatment of the long-range periodic interactions that underpin the physics of extended systems. We apply this graph-theoretical treatment to study PFOA on the surface of water as a model system for the study of weak interactions seen in the wide range of surface interactions and reactions. The approach divides a system into a set of vertices, that are then connected through edges, faces, and higher order graph theoretic objects known as simplexes, to represent a collection of locally interacting subsystems. These subsystems are then used to construct ab initio molecular dynamics simulations and for computing multidimensional potential energy surfaces. To further improve the computational efficiency of our graph theoretic fragmentation method, we use a recently developed transfer learning protocol to construct the full system potential energy from a family of neural networks each designed to accurately model the behavior of individual simplexes. We use a unique multidimensional clustering algorithm, based on the k-means clustering methodology, to define our training space for each separate simplex. These models are used to extrapolate the energies for molecular dynamics trajectories at PFOA water interfaces, at less than one-tenth the cost as compared to a regular molecular fragmentation-based dynamics calculation with excellent agreement with couple cluster level of full system potential energies.

Diclique digraphs

Given a digraph D, a diclique is a maximal pair of vertex sets (X,Y) satisfying x∈X,y∈Y⟹x→y. The diclique digraph of D, K⃗(D), has the dicliques of D as vertices two of them being adjacent, (X,Y)→(X′,Y′), if and only if Y∩X′≠∅. Iterated diclique digraphs are defined by K⃗0(D)=D and K⃗n(D)=K⃗(K⃗n−1(D)). Here we study the diclique operatorK⃗ from the perspective of graph dynamics (Prisner, 1995) and hence we are interested in K⃗-divergence (|K⃗n(D)|→∞), K⃗-convergence (K⃗n(D)≅K⃗m(D) for some n<m), K⃗-invariance (K⃗(D)≅D) and similar properties. In particular, we provide here the first examples of simple digraphs which are K⃗-divergent.

Occupancy probabilities in superintegrable bosonic networks

We associate a bosonic network to each complete bipartite graph. A Hamiltonian is defined with hopping terms on the edges of the graph, and a global interaction term depending on the vertex sets. For a generic graph this Hamiltonian is superintegrable, and we derive a Bethe Ansatz solution for the energy and eigenstates. These results provide the means to investigate the quantum dynamics and allow for the computation of occupancy probabilities in certain regimes. We use these results to gain an understanding of entanglement evolution in the network.

Rainbow Hamilton cycles in random geometric graphs

AbstractLet be chosen independently and uniformly at random from the unit ‐dimensional cube . Let be given and let . The random geometric graph has vertex set and an edge whenever . We show that if each edge of is colored independently from one of colors and has the smallest value such that has minimum degree at least two, then contains a rainbow Hamilton cycle asymptotically almost surely.

The Geometric–Arithmetic index of trees with a given total domination number

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. In Vukičević and Furtula (2009), Vukičević defined a new topological index, named geometric–arithmetic index of a graph G and denoted by GA(G), as GAG=∑uv∈E2dudvdu+dv,where du and dv denote the degrees of vertices u and v, respectively. We obtain an upper bound for the geometric–arithmetic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound. Additionally, by a known relation between the geometric–arithmetic and arithmetic–geometric indices, we get a new lower bound for the arithmetic–geometric index.

More Ramsey theory for highly connected monochromatic subgraphs

Abstract An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$ , • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $ . Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$ . To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

In this paper, we are interested in algorithms that take in input an arbitrary graph G, and that enumerate in output all the (inclusion-wise) maximal “subgraphs” of G which fulfil a given property Π. All over this paper, we study several different properties Π, and the notion of subgraph under consideration (induced or not) will vary from a result to another.More precisely, we present efficient algorithms to list all maximal split subgraphs, maximal induced cographs and maximal threshold graphs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal stable sets of an auxiliary graph. For cographs and threshold graphs, the algorithms rely on a framework recently introduced by Conte & Uno (2019) called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every non-trivial hereditary property Π. We extend the hardness result to some specific edge version of the extension problem.

Minimal spread of integral circulant graphs

The integral circulant graph ICGn(D) has the vertex set Zn={0,1,2,…,n−1} and vertices a and b are adjacent if gcd(a−b,n)∈D, where D⊆{d:d∣n,1≤d<n}. In this paper we prove that the minimal value of the spread of the integral circulant graphs ICGn(D) of a given order n with its prime factorization p1α1⋯pkαk, is equal to ∑i=1kpiαi−1 or ∑i=1kpiαi, depending on whether n=2⋅3⋅p3α3⋯pkαk or not, respectively. Furthermore, we completely characterize all graphs whose minimal spread attains the mentioned values in the class of connected integral circulant graphs of a given order.

Minimal toughness in special graph classes

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

Generalized cut trees for edge-connectivity

We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following:•A pair of vertices {v,w} of a graph G is pendant if λ(v,w)=min⁡{d(v),d(w)}. Mader showed in 1974 that every simple graph with minimum degree δ contains at least δ(δ+1)/2 pendant pairs. We improve this lower bound to δn/24 for every simple graph G on n vertices with δ≥5 or λ≥4 or vertex connectivity κ≥3, and show that this is optimal up to a constant factor with regard to every parameter.•Every simple graph G satisfying δ>0 has O(n/δ)δ-edge-connected components. Moreover, every simple graph G that satisfies 0<λ<δ has O((n/δ)2) cuts of size less than min⁡{32λ,δ}, and O((n/δ)⌊2α⌋) cuts of size at most min⁡{α⋅λ,δ−1} for any given real number α≥1.•A cut is trivial if it or its complement in V(G) is a singleton. We provide an alternative proof of the following recent result of Lo et al.: Given a simple graph G on n vertices that satisfies δ>0, we can compute vertex subsets of G in near-linear time such that contracting these vertex subsets separately preserves all non-trivial min-cuts of G and leaves a graph having O(n/δ) vertices and O(n) edges.

On the Parallel Parameterized Complexity of MaxSAT Variants

In the maximum satisfiability problem (max-sat) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of max-sat and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee. For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for max-2sat (known as almost-2sat). The difficulty in solving almost-2sat in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of max-sat parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input’s incidence graph. While max-sat is fixedparameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment – in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

Resolvability and Convexity Properties in the Sierpiński Product of Graphs

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Prime vertex-minors of a prime graph

A graph is prime if it does not admit a partition (A,B) of its vertex set such that min{|A|,|B|}≥2 and the rank of the A×B submatrix of its adjacency matrix is at most 1. A vertex v of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at v result in prime graphs.In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph G with at least six vertices and a vertex x, there is a vertex v≠x such that G∖v or G∗v∖v is prime, unless x is adjacent to all other vertices and G is isomorphic to a particular graph on odd number of vertices.Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.