Internal Vertices Research Articles

Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V ( G ) , let I ( S ) = ∪ u , v ∈ S I [ u , v ] . If I ( S ) = V ( G ) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g ( G ) . In this paper, the exact geodetic numbers of the product graphs T × K m and T ° K ¯ m are obtained, where T is a tree, K ¯ m denotes the complement of the complete graph K m and, × and ° denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

On damping a control system with global aftereffect on quantum graphs: Stochastic interpretation

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff‐type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first‐order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self‐adjoint boundary value problem on the tree for some second‐order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.

Mutual-visibility and general position in double graphs and in Mycielskians

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of vertices that can be selected such that every pair of vertices in the collection has a shortest path between them with no vertex from the collection as an internal vertex. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total and the outer mutual-visibility number of base graphs. Several exact values are also determined, in particular the mutual-visibility number of the double graphs and of the Mycielskian of cycles.

On the existence of funneled orientations for classes of rooted phylogenetic networks

Recently, there has been a growing interest in the relationships between unrooted and rooted phylogenetic networks. In this context, a natural question to ask is if an unrooted phylogenetic network U can be oriented as a rooted phylogenetic network such that the latter satisfies certain structural properties. In a recent preprint, Bulteau et al. claim that it is NP-hard to decide if U has a funneled (resp. funneled tree-child) orientation, for when the internal vertices of U have degree at most 5. Unfortunately, the proof of their funneled tree-child result appears to be incorrect. In this paper, we show that, despite their incorrect proof, it is NP-hard to decide if U has a funneled tree-child orientation even if each internal vertex has degree 5 and that NP-hardness remains for other popular classes of rooted phylogenetic networks such as funneled normal and funneled reticulation-visible. Additionally, our results hold regardless of whether U is rooted at an existing vertex or by subdividing an edge with the root.

Minimising the eccentricity index in trees with given segment sequence

The eccentricity index of a graph is the sum of the eccentricities of all vertices in the graph. A segment S of a tree T is a subgraph isomorphic to a non-trivial path where all internal vertices have degree 2 in T and the end-vertices are branching vertices or leaves. The segment sequence of a tree is the sequence of the lengths of all segments in a tree. In this paper, we characterise trees of a given segment sequence with minimum eccentricity index. We show that in the family of trees of a given segment sequence, the tree with minimum eccentricity index is the unique ‘starlike’ tree (a tree with one branching vertex).

Fiber-reinforced shotcrete lining for stabilizing rock blocks around underground cavities

Fiber-reinforced shotcrete is a high-performance material that presents some special characteristics, which can provide some suitable applications in the excavation of underground cavities. The presence of fibers induces an increase in the tensile strength, flexural strength and shear strength in the concrete, as well as allowing a ductile rather than brittle type of behavior. It can be used to create a lining of the underground cavity that allows the stabilization of rock blocks that show a tendency to slip or fall (from the side walls or from the crown area, respectively). In this work, some full-scale tests on the fiber-reinforced shotcrete lining are presented. From these tests, it was possible to measure the behavior of this material when it is loaded locally: it is the same type of action produced by the rock block when it is held back from falling or slipping. The results obtained have allowed to characterize this type of material from a mechanical point of view. A subsequent detailed analysis of the stability of rock blocks surrounding an underground cavity permitted to determine the static stabilizing contribution offered by the fiber-reinforced shotcrete lining, leading to the definition of the minimum thickness required, in relation to the type of block that is present (shape and size). It was possible to predict how a lining thickness of about 3.5 cm is able to stabilize (just 15 min after its spraying) rock blocks with an exposed surface area of up to 10 m2 and a distance of the internal vertex from the border of the cavity of up to 3 m

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g) is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric dg induced on the boundary ∂M? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in R3 can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

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.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

Monophonic graphoidal covering number of corona product graphs

In a graph G, a chordless path is called a monophonic path. A collection ψm of monophonic paths in G is called a monophonic graphoidal cover of G if every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The monophonic graphoidal covering number ηm(G) of G is the minimum cardinality of a monophonic graphoidal cover of G. In this paper, we find the monophonic graphoidal covering number of corona product of some standard graphs.

Lower (total) mutual-visibility number in graphs

Given a graph G, a set X of vertices in G satisfying that between every two vertices in X (respectively, in G) there is a shortest path whose internal vertices are not in X is a mutual-visibility (respectively, total mutual-visibility) set in G. The cardinality of a largest (total) mutual-visibility set in G is known under the name (total) mutual-visibility number, and has been studied in several recent works.In this paper, we propose two lower variants of these concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in G, and denote them by μ−(G) and μt−(G), respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph G, we prove that both differences μ−(G)−μt−(G) and μt−(G)−μ−(G) can be arbitrarily large. We characterize graphs G with some small values of μ−(G) and μt−(G), and prove a useful tool called the Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with the Bollobás-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to μt−(G).

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

The rainbow vertex connection number of ladder graphs and Roach graphs

A vertex-coloured graph G is said to be rainbow vertex-connected, if every two vertices of G are connected by a path whose internal vertices have distinct colours. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colours that are needed to make G, a rainbow vertex-connected. This study focuses on deriving formulas for the rainbow vertex connectivity number of a simple ladder graph and a roach graph.

Determining triangulations and quadrangulations by boundary distances

We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. However, we show that a natural conjecture for a “mixed” version of the two results is not true.

The product structure of squaregraphs

AbstractA squaregraph is a plane graph in which each internal face is a 4‐cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semistrong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that “outerplanar graph” cannot be replaced by “forest”.

On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs

Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PDF

Edge even graceful labeling of [formula omitted] and [formula omitted

For a graph G(V,E), with EG=m and VG=n, the edge even graceful labeling is a bijection f:EG→{2,4,6,⋯,2m} such that, each vertex vi is assigned by the color ∑fvivjmod2r. Where r=max{n,m},vj∈Nvi,vivj∈EG, the resulting vertex label is distinct. In this article the graphs TPn, a rooted tree T with root vertex of degree d-1 and internal vertices of odd degree d, Frn3 and Fn=K1+Pn, Wn proved as an edge even graceful graph.

On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

Shape reconstruction for undetectable regions of abdominal organs based on a graph convolutional network

Although computed tomography (CT) and magnetic resonance imaging (MRI) produce high-resolution images, during surgery or radiotherapy, only low-resolution cone-beam CT and low-dimensional X-ray images are usually obtained. Furthermore, because the duodenum and stomach are filled with air, it may be hard to accurately segment their contours, even on high-resolution images. Although denoising and image enhancement can be used to assist in the segmentation and reconstruction of 3D images, some regions of organs may remain difficult to detect. To approach this issue, we propose a graph convolutional network (GCN) to reconstruct organs that are hard to detect on medical images. The method uses a framework to transform an initial template into an estimated shape of the patient’s organs based on previous information attainted from different detectable organs from different patients. To overcome the inaccurate estimation results of triangular surface mesh data with the GCN, we propose a method to add internal vertices and edges, which has the effect of improving the calculation accuracy by enhancing the topological structure of the mesh data. In this study, the organ data of 124 patients who underwent pancreatic tumor treatment were used to verify the predictive effect of the proposed method. The 3D organ contours were defined by board-certified radiation oncologists and used as the ground truth surface meshes. For 100 randomly selected detectable feature vertices, the average Euclidean distance error of the liver was 3.72 mm. The topological enhancement proposed in this study reduced the prediction error of the liver by an average of 0.84 mm.

Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

We study reconfiguration of simple Hamiltonian cycles in a rectangular grid graph [Formula: see text], where the Hamiltonian cycle in each step of the reconfiguration connects every internal vertex of [Formula: see text] to a boundary vertex by a single straight line segment. We introduce two operations, flip and transpose, which are local to the grid. We show that any simple cycle of [Formula: see text] can be reconfigured to any other simple cycle of [Formula: see text] using [Formula: see text] flip and transpose operations. Our result proves that the simple Hamiltonian cycle graph [Formula: see text] is connected with respect to those two operations and has diameter [Formula: see text].