Pair Of Vertices Research Articles

Mutual-visibility in strong products of graphs via total mutual-visibility

Let G be a graph and X⊆V(G). Then X is a mutual-visibility set if each pair of vertices from X is connected by a geodesic with no internal vertex in X. The mutual-visibility number μ(G) of G is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.

Minimum length word-representants of graph products

A graph G=(V,E) is said to be word-representable if a word w can be formed using the letters of the alphabet V such that for every pair of vertices x and y, x≠y, xy∈E if and only if x and y alternate in w. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles. It is known that Cartesian and rooted products preserve word-representability. Moreover, using occurrence-based functions, Broere constructed a uniform word representing the Cartesian product of G and Kn.In this paper, we study the minimum length of word-representants for Cartesian and rooted products using morphism and occurrence-based functions, respectively. Also, we solve an open problem posed by Broere in his master thesis. This problem asks to construct a word for the Cartesian product of two arbitrary word-representable graphs.

The number of 1-nearly independent vertex subsets

Let G be a graph with vertex set V(G) and edge set E(G). A subset I of V(G) is an independent vertex subset if no two vertices in I are adjacent in G. We study the number, σ 1(G), of all subsets of V(G) that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of σ 1 are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on σ 1 for graphs of order n. We deduce as a corollary that the star K 1,n−1 (the tree with degree sequence (n − 1, 1, . , 1)) is the n-vertex tree with smallest σ 1, while it is well known that K1,n−1 is the n-vertex tree with largest number of independent subsets.

A Note on the Distribution of the Extreme Degrees of a Random Graph via the Stein-Chen Method

We offer an alternative proof, using the Stein-Chen method, of Bollobás’ theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.

Morphology of the scaphotrapeziotrapezoid joint: A multi-domain statistical shape modeling approach.

The scaphotrapeziotrapezoid (STT) joint is involved in load transmission between the wrist and thumb. A quantitative description of baseline STT joint morphometrics is needed to capture the variation of normal anatomy as well as to guide staging of osteoarthritis. Statistical shape modeling (SSM) techniques quantify variations in three-dimensional shapes and relative positions. The objectives of this study are to describe the morphology of the STT joint using a multi-domain SSM. We asked: (1) What are the dominant modes of variation that impact bone and articulation morphology at the STT joint, and (2) what are the morphometrics of SSM-generated STT joints? Thirty adult participants were recruited to a computed tomography study of normal wrist imaging and biomechanics. Segmentations of the carpus were converted to three-dimensional triangular surface meshes. A multi-domain, particle-based entropy system SSM was used to quantify variation in carpal bone shape and position as well as articulation morphology. Articular surface areas and interosseous proximity distributions were calculated between mesh vertex pairs on adjacent bones within distance (2.0 mm) and surface-normal angular (35°) thresholds. In the SSM, the first five modes of variation captured 76.2% of shape variation and contributed to factors such as bone scale, articular geometries, and carpal tilt. Median interosseous proximities-a proxy for joint space-were 1.39 mm (scaphotrapezium), 1.42 mm (scaphotrapezoid), and 0.61 mm (trapeziotrapezoid). This study quantifies morphological and articular variations at the STT joint, presenting a range of normative anatomy. The range of estimated interosseous proximities may guide interpretation of imaging-derived STT joint space.

On the Problems of CF-Connected Graphs for Kl,m,n

A connected graph, G, is Crossing Free-connected (CF-connected) if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete tripartite graph, Kl,m,n, is CF-connected if and only if it does not contain any of the following as a subgraph: K1,2,7, K1,3,5, K1,4,4, K2,2,5, K3,3,3. We examine the idea that K1,2,7, K1,3,5, K1,4,4, and K2,2,5 are the first non-CF-connected complete tripartite graphs. The CF-connectedness of Kl,m,n with l,m,n≥3 is dependent on the knowledge of crossing numbers of K3,3,n. In this paper, we prove various results that support this conjecture.

Further results on the radio number for some construction of the path, complete, and complete bipartite graphs

Let G be a connected graph with diam(G). Then d(u,v) indicates the distance of u and v in G. For any pair of distinct vertices u,v of G, mapping from c:V(G)→N such that dG(u,v)+|c(u)−c(v)|≥diam(G)+1. The maximum label assigned to any vertex of G under a radio labeling c is known as the span of c. The radio number rn(G) of G is defined as the minimum span among all possible radio labelings of G. This paper aims to determine the radio number rn(c) for specific constructed families of graphs with diameter 3, such as P22(N(m,n)), K3,m(Knc), K3,m(nP3), K3,3(2K1,n), and K5(F3n).

The edge fault-tolerance about the strong Menger edge-connectivity of order [formula omitted] among hamming graph

Let r≥0 be a fixed integer. If F is an edge set of a connected graph G satisfying the minimum degree of G−F being at least r, then F is a conditional faulty edge set of order r. The graph G is called F-strongly Menger-edge-connected if each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} edge-disjoint paths in G−F, where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. A graph G is m-strongly Menger-edge-connected of order r if G is F-strongly Menger-edge-connected of order r for every F⊂E(G) with |F|≤m and F is a conditional edge fault set of order r, and the maximum value of m is written as smλr(G). Hamming graph KLn has been widely concerned by researchers due to its excellent properties such as good connectivity, scalability, symmetry and iterativity. This paper considers various sufficient conditions of KLn to be F-strongly Menger-edge-connected of order r and determine the exact value of smλ(L−1)r(KLn) by studying the edge-disjoint paths in KLn with edge faults, smλ(L−1)r(KLn)=(L−1)Lr(n−r)−(L−1)n, where 0≤r≤n−2 and n≥3.

Discovering the signal subgraph: An iterative screening approach on graphs

Supervised learning on graphs is a challenging task due to the high dimensionality and inherent structural dependencies in the data, where each edge depends on a pair of vertices. Existing conventional methods are designed for standard Euclidean data and do not account for the structural information inherent in graphs. In this paper, we propose an iterative vertex screening method to achieve dimension reduction across multiple graph datasets with matched vertex sets and associated graph attributes. Our method aims to identify a signal subgraph to provide a more concise representation of the full graphs, potentially benefiting subsequent vertex classification tasks. The method screens the rows and columns of the adjacency matrix concurrently and stops when the resulting distance correlation is maximized. We establish the theoretical foundation of our method by proving that it estimates the true signal subgraph with high probability. Additionally, we establish the convergence rate of classification error under the Erdos-Renyi random graph model and prove that the subsequent classification can be asymptotically optimal, outperforming the entire graph under high-dimensional conditions. Our method is evaluated on various simulated datasets and real-world human and murine graphs derived from functional and structural magnetic resonance images. The results demonstrate its excellent performance in estimating the ground-truth signal subgraph and achieving superior classification accuracy.

Approximate Distance Sensitivity Oracles in Subquadratic Space

An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $\sigma \ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $\sigma$-approximation of the $s$-$t$-distance in $G-F$. We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $\sigma \ge 3$. We present, for any constant $f \ge 1$ and $\alpha \in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\frac{\alpha}{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^\alpha/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\frac{\alpha}{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$. Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$.

RNA-clique: a method for computing genetic distances from RNA-seq data

BackgroundAlthough RNA-seq data are traditionally used for quantifying gene expression levels, the same data could be useful in an integrated approach to compute genetic distances as well. Challenges to using mRNA sequences for computing genetic distances include the relatively high conservation of coding sequences and the presence of paralogous and, in some species, homeologous genes.ResultsWe developed a new computational method, RNA-clique, for calculating genetic distances using assembled RNA-seq data and assessed the efficacy of the method using biological and simulated data. The method employs reciprocal BLASTn followed by graph-based filtering to ensure that only orthologous genes are compared. Each vertex in the graph constructed for filtering represents a gene in a specific sample under comparison, and an edge connects a pair of vertices if the genes they represent are best matches for each other in their respective samples. The distance computation is a function of the BLAST alignment statistics and the constructed graph and incorporates only those genes that are present in some complete connected component of this graph. As a biological testbed we used RNA-seq data of tall fescue (Lolium arundinaceum), an allohexaploid plant (2n=14Gb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n = 14\ ext { Gb}$$\\end{document}), and bluehead wrasse (Thalassoma bifasciatum), a teleost fish. RNA-clique reliably distinguished individual tall fescue plants by genotype and distinguished bluehead wrasse RNA-seq samples by individual. In tests with simulated RNA-seq data, the ground truth phylogeny was accurately recovered from the computed distances. Moreover, tests of the algorithm parameters indicated that, even with stringent filtering for orthologs, sufficient sequence data were retained for the distance computations. Although comparisons with an alternative method revealed that RNA-clique has relatively high time and memory requirements, the comparisons also showed that RNA-clique’s results were at least as reliable as the alternative’s for tall fescue data and were much more reliable for the bluehead wrasse data.ConclusionResults of this work indicate that RNA-clique works well as a way of deriving genetic distances from RNA-seq data, thus providing a methodological integration of functional and genetic diversity studies.

On the computational complexity of the strong geodetic recognition problem

A vertex set $S$ of a graph $G$ is a strong geodetic set if it is possible to choose exactly one shortest path for each pair of vertices in $S$, and cover all the remaining vertices of the graph with these paths. In the \textsc{Strong Geodetic} problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be \NP-complete for general graphs. In this work we introduce the \textsc{Strong Geodetic Recognition} problem (SGR), which consists in determining whether a given vertex set~$S \subseteq V(G)$ is strong geodetic. We demonstrate that this version is \NP-complete. We investigate and compare the computational complexity of both decision problems restricted to some graph classes, deriving polynomial-time algorithms, \NP-completeness proofs, and initial parameterized complexity results, including an answer to an open question in the literature for the complexity of SG for chordal graphs.

Distributed Shortest Distance Labeling on Large-Scale Graphs

Distance labeling approaches are widely adopted to speed up the shortest-distance query performance. Due to the explosive growth of data graphs, a single machine can hardly satisfy the requirements of both computational power and memory capacity, which causes an urgent need for efficient distributed methods. As the graph is distributed across different machines, it is inevitable to frequently exchange messages among different machines when deploying the existing centralized distance labeling methods on the distributed environment, thereby producing serious communication costs and weakening the scalability. To alleviate this problem, we design a distributed hop-based index DH-Index , which is designed based on a newly proposed boundary graph structure and restricts the index-based hop number of each connected vertex pair within 4 hops. In addition, we propose a hierarchical algorithm to accelerate the index construction and reduce the communication cost. Furthermore, a bidirectional searching strategy is proposed to efficiently resolve the query tasks based on DH-Index. The comprehensive experimental results on eight real-world graphs demonstrate that DH-Index achieves up to 65.5× and 3 orders of magnitude speedup than the existing methods in indexing time and query performance respectively, and exhibits superior capabilities on memory space, communication cost, and scalability.

Generalisations of matrix partitions: Complexity and obstructions

A trigraph is a graph where each pair of vertices is labelled either 0 (a non-arc), 1 (an arc) or ⋆ (both an arc and a non-arc). In a series of papers, Hell and co-authors (see for instance [P. Hell, 2014 [21]]) proposed to study the complexity of homomorphisms from graphs to trigraphs, called Matrix Partition Problems, where arcs and non-arcs can be both mapped to ⋆-arcs, while a non-arc cannot be mapped to an arc, and vice-versa. Even though Matrix Partition Problems are generalisations of Constraint Satisfaction Problems (CSPs), they share with them the property of being “intrinsically” combinatorial. So, the question of a possible P-time vs NP-complete dichotomy is a very natural one and was raised in Hell et al.'s papers. We propose a generalisation of Matrix Partition Problems to relational structures and study them with respect to the question of a dichotomy. We first show that trigraph homomorphisms and Matrix Partition Problems are P-time equivalent, and then prove that one can also restrict (with respect to having a dichotomy) to relational structures with a single relation. Failing in proving that Matrix Partition Problems on directed graphs are not P-time equivalent to Matrix Partitions on relational structures, we give some evidence that it might be unlikely by formalising the reductions used in the case of CSPs and by showing that such reductions cannot work for the case of Matrix Partition Problems. We then turn our attention to Matrix Partition problems that can be described by finite sets of (induced-subgraph) obstructions. We show, in particular, that any such problem has finitely many minimal obstructions if and only if it has finite duality. We conclude by showing that on trees (seen as trigraphs) it is NP-complete to decide whether a given tree has a homomorphism to another input trigraph. The latter shows a notable difference on tractability between CSP and Matrix Partition Problems as it is well-known that CSP is tractable on the class of trees.

Monitoring Edge-Geodetic Numbers of Mycielskian Graph Classes

Let [Formula: see text] be a graph and [Formula: see text] be the vertex set of [Formula: see text]. If any edge appears in the unique shortest path of at least one vertex pair in set [Formula: see text], then [Formula: see text] is defined as the monitoring edge-geodetic set (MEG-set for short). We denote by [Formula: see text] the size of a smallest MEG-set of [Formula: see text]. In this paper, we studied the monitoring edge-geodetic numbers of Mycielski graph classes.

On the general position number of the k-th power graphs

A set is called a general position set of a graph G if any triple set V 0 of R is non-geodesic, that is, three elements of V 0 cannot lie on the same geodesic of G. The general position number gp(G) of a graph G is the cardinality of a largest general position set of G. The k-th power, Gk , of a graph G is obtained from G by adding a new edge between each pair of vertices with a distance of at most k in G. In this paper we establish the bounds for gp(Gk ) and and give an explicit formula of . Also we consider the relation between gp(G 2) and gp(G). We deduce a formula of and provide the upper bound on gp(G 2)−gp(G) for general block graphs G. Moreover, we determine the gp-number for the square of trees.

Spanning even trees of graphs

AbstractA tree is said to be even if all pairs of vertices of degree one in are joined by a path of even length. We conjecture that every ‐regular nonbipartite connected graph has a spanning even tree and verify this conjecture when has a 2‐factor. Well‐known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when is odd and has at least bridges. We investigate this case further and propose some related conjectures.

On the algebraic connectivity of some token graphs

The k-token graph Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.

The Euler Sombor index of a graph

AbstractA novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors, which is the perimeter of an ellipse whose focal points represent the degree‐point and its dual‐point of a pair of adjacent vertices in a graph . We first analyze the chemical applicability of this elliptic Sombor index, and then establish its mathematical properties and determine the extremal values for this elliptic Sombor index among all (molecular) trees and characterize the corresponding extremal graphs.

Arc-disjoint out- and in-branchings in compositions of digraphs

An out-branching Bu+ (in-branching Bu−) in a digraph D is a connected spanning subdigraph of D in which every vertex except the vertex u, called the root, has in-degree (out-degree) one. A good(u,v)-pair in D is a pair of branchings Bu+,Bv− which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is semicomplete if it has no pair of non-adjacent vertices. A semicomplete composition is any digraph D which is obtained from a semicomplete digraph S by substituting an arbitrary digraph Hx for each vertex x of S.Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good (u,v)-pair, where u,v are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph D and vertices u,v of D, either produces a good (u,v)-pair in D or a certificate that D has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good (u,v)-pair for given vertices u,v. Our solution implies that the problem of deciding the existence of a good (u,v)-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good (u,v)-pair and finding one when it exists for digraphs that are compositions of transitive digraphs. Combining these two results we obtain a polynomial algorithm for deciding whether a given quasi-transitive digraph D has a good (u,v)-pair for given vertices u,v of D. This proves a conjecture of Bang-Jensen and Gutin from 1998.