Incident Edges Research Articles

Conjectures About Wheels Without One Edge with Paths and Cycles

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of this paper is to give the crossing numbers of the join products G*+Pn and G*+Cn for the connected graph G* obtained by removing one edge (incident with the dominating vertex) from the wheel W5 on six vertices, and where Pn and Cn are paths and cycles on n vertices, respectively. Finally, we also introduce four new conjectures concerning crossing numbers of the join products of Pn and Cn with Wm∖e obtained by removing one edge (of both possible types) from the wheel Wm on m+1 vertices.

Illuminating a Graph

We introduce a two-player game where the goal is to illuminate all edges of a graph. At each step the first player, called Illuminator, taps a vertex. The second player, called Adversary, reveals the edges incident with that vertex (consistent with the edges incident with the already tapped vertices). Illuminator tries to minimize the taps needed, and the value of the game is the number of taps needed with optimal play. We provide bounds on the value in trees and general graphs. In particular, we show that the value for the path on n vertices is 2 3 n + O ( 1 ) , and there is a constant ε > 0 such that for every caterpillar on n vertices, the value is at most ( 1 – ε ) n + 1 .

Compatible powers of Hamilton cycles in dense graphs

AbstractThe notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.

Splitting Plane Graphs to Outerplanarity

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is NP-hard.Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove NP-completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Additionally, we show upper and lower bounds for certain families of maximal planar graphs. Finally, we provide a SAT formulation for the problem, and evaluate it on a small benchmark.

Pseudo-Spectral Spatial Feature Extraction and Enhanced Fusion Image for Efficient Meter-Sized Lunar Impact Crater Automatic Detection in Digital Orthophoto Map.

Impact craters are crucial for our understanding of planetary resources, geological ages, and the history of evolution. We designed a novel pseudo-spectral spatial feature extraction and enhanced fusion (PSEF) method with the YOLO network to address the problems encountered during the detection of the numerous and densely distributed meter-sized impact craters on the lunar surface. The illumination incidence edge features, isotropic edge features, and eigen frequency features are extracted by Sobel filtering, LoG filtering, and frequency domain bandpass filtering, respectively. Then, the PSEF images are created by pseudo-spectral spatial techniques to preserve additional details from the original DOM data. Moreover, we conducted experiments using the DES method to optimize the post-processing parameters of the models, thereby determining the parameter ranges for practical deployment. Compared with the Basal model, the PSEF model exhibited superior performance, as indicated by multiple measurement metrics, including the precision, recall, F1-score, mAP, and robustness, etc. Additionally, a statistical analysis of the error metrics of the predicted bounding boxes shows that the PSEF model performance is excellent in predicting the size, shape, and location of impact craters. These advancements offer a more accurate and consistent method to detect the meter-sized craters on planetary surfaces, providing crucial support for the exploration and study of celestial bodies in our solar system.

BrepGen: A B-rep Generative Diffusion Model with Structured Latent Geometry

This paper presents BrepGen , a diffusion-based generative approach that directly outputs a Boundary representation (B-rep) Computer-Aided Design (CAD) model. BrepGen represents a B-rep model as a novel structured latent geometry in a hierarchical tree. With the root node representing a whole CAD solid, each element of a B-rep model (i.e., a face, an edge, or a vertex) progressively turns into a child-node from top to bottom. B-rep geometry information goes into the nodes as the global bounding box of each primitive along with a latent code describing the local geometric shape. The B-rep topology information is implicitly represented by node duplication. When two faces share an edge, the edge curve will appear twice in the tree, and a T-junction vertex with three incident edges appears six times in the tree with identical node features. Starting from the root and progressing to the leaf, BrepGen employs Transformer-based diffusion models to sequentially denoise node features while duplicated nodes are detected and merged, recovering the B-Rep topology information. Extensive experiments show that BrepGen advances the task of CAD B-rep generation, surpassing existing methods on various benchmarks. Results on our newly collected furniture dataset further showcase its exceptional capability in generating complicated geometry. While previous methods were limited to generating simple prismatic shapes, BrepGen incorporates free-form and doubly-curved surfaces for the first time. Additional applications of BrepGen include CAD autocomplete and design interpolation. The code, pretrained models, and dataset are available at https://github.com/samxuxiang/BrepGen.

Total Labelings of Graphs with Prescribed Weights

The total labeling of a graph G = ( V , E ) is a bijection from the union of the vertex set and the edge set of G to the set { 1 , 2 , … , | V ( G ) | + | E ( G ) | } . The edge-weight of an edge under a total labeling is the sum of the label of the edge and the labels of the end vertices of that edge. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-magic or vertex-magic when all the edge-weights or all the vertex-weights are the same, respectively. When all the edge-weights or all the vertex-weights are different then a total labeling is called edge-antimagic or vertex-antimagic total, respectively. In this paper we deal with the problem of finding a~total labeling of some classes of graphs that is simultaneously vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic, respectively. We show several results for stars, paths and cycles.

On Strong Regal and Strong Royal Colorings

Two colorings have been introduced recently where an unrestricted coloring c assigns nonempty subsets of [ k ] = { 1 , … , k } to the edges of a (connected) graph G and gives rise to a vertex-distinguishing vertex coloring by means of set operations. If each vertex color is obtained from the union of the incident edge colors, then c is referred to as a strong royal coloring. If each vertex color is obtained from the intersection of the incident edge colors, then c is referred to as a strong regal coloring. The minimum values of k for which a graph G has such colorings are referred to as the strong royal index of G and the strong regal index of G respectively. If the induced vertex coloring is neighbor distinguishing, then we refer to such edge colorings as royal and regal colorings. The royal chromatic number of a graph involves minimizing the number of vertex colors in an induced vertex coloring obtained from a royal coloring. In this paper, we provide new results related to these two coloring concepts and establish a connection between the corresponding chromatic parameters. In addition, we establish the royal chromatic number for paths and cycles.

On the power graphs of semigroups of homogeneous elements of graded semisimple Artinian rings

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G ( H R ) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * ( H R ) a graph obtained from G ( H R ) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * ( R e ) denotes a subgraph of G * ( H R ) , induced by R e * . In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t . Also, let R be semisimple Artinian. We prove that if G * ( R e ) is connected for every nonzero ring component Re of R, then the connected components of G * ( H R ) are precisely the graphs G * ( R e ) .

Palindromic Antimagic Labeling of Products of Paw and Banner Graph

This article presents a novel classification of Antimagic labeling referred to as Palindromic antimagic. Palindromic Antimagic labeling pertains to the assignment of palindromic numbers {℘1, ℘2, ℘3 . . . ℘q} to the set of edges of a graph G = (V, E), where G consists of p vertices and q edges. This labeling is characterized by being invertible, meaning that each edge is uniquely associated with a palindromic number. Additionally, it involves an injective mapping of vertex labeling, where the sum of incident edges for any given vertex is distinct from one another. Inthis study, we examine the palindromic antimagic labeling of Cartesian and Tensor product of certain Tadpole graphs, such as the paw and banner graph.

Perturbation Results for Distance-edge-monitoring Numbers*

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph G = (V(G), E(G)), a set M ⊆ V(G) is a distance-edge-monitoring set if for every edge e ∈ E(G), there is a vertex x ∈ M and a vertex y ∈ V(G) such that the edge e belongs to all shortest paths between x and y. The smallest size of such a set in G is denoted by dem(G). Denoted by G – e (resp. G\u) the subgraph of G obtained by removing the edge e from G (resp. a vertex u together with all its incident edges from G). In this paper, we first show that dem(G – e) – dem(G) ≤ 2 for any graph G and edge e ∈ E(G). Moreover, the bound is sharp. Next, we construct two graphs G and H to show that dem(G) – dem(G\u) and dem(H \ v) – dem(H) can be arbitrarily large, where u ∈ V(G) and v ∈ V(H). We also study the relation between dem(H) and dem(G), where H is a subgraph of G. In the end, we give an algorithm to judge whether the distance-edge-monitoring set still remain in the resulting graph when any edge of a graph G is deleted.

Bakry–Émery–Ricci curvature: an alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

Abstract The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. Various types of such Ricci curvatures capture different aspects of network geometry. In the present work, we investigate Bakry–Émery–Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Furthermore, the curvature distributions are different for different types of model networks. We observe a high positive correlation between Bakry–Émery–Ricci and both Forman–Ricci and Ollivier–Ricci curvature, and in particular with the augmented version of Forman–Ricci curvature while comparing them for both model and real-world networks. We investigate the correlation of Bakry–Émery–Ricci curvature with degree, clustering coefficient and vertex centrality measures. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. Additionally, for Forman–Ricci, Augmented Forman–Ricci and Ollivier–Ricci curvature, we compare the robustness of the networks by comparing the sum of the incident edges and the minimum of the incident edges as vertex measures and find that the sum identifies vertices that are important for maintaining the connectivity of the network. The computational time for Bakry–Émery–Ricci curvature is shorter than that required for Ollivier–Ricci curvature but higher than for Augmented Forman–Ricci curvature. We therefore conclude that for empirical network analysis, the latter is the tool of choice.

Edges incident with a vertex of degree greater than four and the number of contractible edges in a 4-connected graph

Let G be a 4-connected graph, and let Ec(G) denote the set of 4-contractible edges of G and V≥5(G) denote the set of vertices of G whose degree is greater than 4. Under this notation, we show that |Ec(G)|≥(1/26)∑u∈V≥5(G)degG(u).

Edge incident 2-edge coloring of graphs

The edge incident 2-edge coloring of a graph [Formula: see text] is an edge coloring of the graph [Formula: see text] such that not more than two colors are assigned to the edges incident to an edge [Formula: see text] in [Formula: see text]. In other words, for every edge [Formula: see text] in [Formula: see text], the edge [Formula: see text] and all the edges that are incident to the edge [Formula: see text] is in at most two different color classes. The edge incident 2-edge coloring number [Formula: see text] is the maximum number of colors in any edge incident 2-edge coloring of [Formula: see text]. The main objective of this paper is to study the edge incident 2-edge coloring concept and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds.

A note on [formula omitted]-coloring and [formula omitted]-coloring 4-regular graphs

Let ∂H(u) be the set of edges incident with a vertex u in the graph H. We say that a graph G is H-colorable if there exist total functions f:E(G)→E(H) and g:V(G)→V(H) such that f is a proper edge-coloring of G and for each vertex u∈V(G) we have f(∂G(u))=∂H(g(u)). Let X¯ be the graph obtained by adding three parallel edges between two degree one vertices of the graph K1,4. Let Aˆ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree two vertex and the two adjacent degree three vertices. Malnegro and Ozeki (2024) [7] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition admits an X¯-coloring or an Aˆ-coloring and whether every 2-connected planar 4-regular graph with an even number of vertices admits such a coloring. Additionally, they conjectured that for every 2-edge-connected simple cubic graph G with an even number of edges, the line graph L(G) is X¯-colorable. In this short note, we discuss two algorithms for deciding whether a graph G is H-colorable. We give negative answers to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.

Results on Extremal Graph Theoretic Questions for Q-Ary Vectors

A 𝑞-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 × 𝑛 matrix with entries from {0, … , 𝑞}, such that each row of the matrix (called a 𝑞-edge) contains exactly two nonzero entries. If 𝐻 is a 𝑞-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of 𝑞-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, 𝑞, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex 𝑞-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, 𝑞, 𝑞 + 1) for even 𝑞 to the case 𝑞 = 2, and determine the asymptotics of ex(𝑛, 𝐶2𝑘+1, 𝑞, 𝑞 + 1).

A new class of disconnected antimagic graphs

A graph G is antimagic if there exists a bijection f from E(G) to {1, 2, . . . , |E(G)|} such that the vertex sums for all vertices of G are distinct, where the vertex sum is defined as the sum of the labels of all incident edges. Hartsfield and Ringel conjectured that every connected graph other than K_2 admits an antimagic labeling. It is still a challenging problem to address antimagicness in the case of disconnected graphs. In this paper, we study antimagicness for the disconnected graph that is constructed as the direct product of a star and a path.

METHOD OF CREATING A MINIMAL SPANNING TREE ON AN ARBITRARY SUBSET OF VERTICES OF A WEIGHTED UNDIRECTED GRAPH

Context. The relevance of the article is determined by the need for further development of models for optimal restoration of the connectivity of network objects that have undergone fragmentation due to emergency situations of various origins. The method proposed in this article solves the problematic situation of minimizing the amount of restoration work (total financial costs) when promptly restoring the connectivity of a selected subset of elements of a network object after its fragmentation.
 The purpose of the study is to develop a method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph to minimize the amount of restoration work and/or total financial costs when promptly restoring the connectivity of elements that have a higher level of importance in the structure of a fragmented network object.
 Method. The developed method is based on the idea of searching for local minima in the structure of a model undirected graph using graph vertices that are not included in the list of base vertices to be united by a minimal spanning tree. When searching for local minima, the concept of an equilateral triangle and a radial structure in such a triangle is used. In this case, there are four types of substructures that provide local minima: first, those with one common base vertex; second, those with two common base vertices; third, those with three common base vertices; fourth, those without common base vertices, located in different parts of the model graph. Those vertices that are not included in the list of basic ones, but through which local minima are ensured, are added to the basic ones. Other vertices (non-basic) along with their incident edges are removed from the structure of the model graph. Then, using one of the well-known methods of forming spanning trees, a minimal spanning tree is formed on the structure obtained in this way, which combines the set of base vertices.
 Results. 1) A method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph has been developed. 2) A set of criteria for determining local minima in the structure of the model graph is proposed. 3) The method has been verified on test problems.
 Conclusions. The theoretical studies and several experiments confirm the efficiency of the developed method. The solutions developed using the developed method are accurate, which makes it possible to recommend it for practical use in determining strategies for restoring the connectivity of fragmented network objects.

Upward planarity testing of biconnected outerplanar DAGs solves partition

We show an O(n)-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to n/2 to the problem of testing whether an n-vertex biconnected outerplanar DAG admits an upward planar drawing. This constitutes the first barrier to the existence of efficient algorithms for testing the upward planarity of DAGs with no large triconnected minor.We also show a result in the opposite direction. Suppose that partitioning a multiset of positive integers into two multisets so that the sum of the integers in each multiset is n/2 can be solved in f(n) time. Let G be an n-vertex biconnected outerplanar DAG and e be an edge incident to the outer face of an outerplanar drawing of G. Then it can be tested in O(f(n)) time whether G admits an upward planar drawing with e on the outer face.

Edge‐minimum saturated k‐planar drawings

AbstractFor a class of drawings of loopless (multi‐)graphs in the plane, a drawing is saturated when the addition of any edge to results in —this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on ‐planar drawings, that is, graphs drawn in the plane where each edge is crossed at most times, and the classes of all ‐planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated ‐planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all ‐vertex saturated ‐planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest ‐vertex saturated ‐planar drawings have edges for any , while if all that is forbidden, the sparsest such drawings have edges for any .