Edge Set Research Articles

Geodetic Hop Dominating Sets in a Graph

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A subset S of vertices of G is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number of G, γhg(G), is the minimum cardinality among all geodetic hop dominating sets in G. Geodetic hop dominating sets in a graph resulting from some binary operations have been characterized. These characterizations have been used to determine some tight bounds for the geodetic hop domination number of each of the graphs considered.

Smaller Embeddings of Partial $k$-Star Decompositions

A $k$-star is a complete bipartite graph $K_{1,k}$. For a graph $G$, a $k$-star decomposition of $G$ is a set of $k$-stars in $G$ whose edge sets partition the edge set of $G$. If we weaken this condition to only demand that each edge of $G$ is in at most one $k$-star, then the resulting object is a partial $k$-star decomposition of $G$. An embedding of a partial $k$-star decomposition $\mathcal{A}$ of a graph $G$ is a partial $k$-star decomposition $\mathcal{B}$ of another graph $H$ such that $\mathcal{A} \subseteq \mathcal{B}$ and $G$ is a subgraph of $H$. This paper considers the problem of when a partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for a given integer $s$. We improve a result of Noble and Richardson, itself an improvement of a result of Hoffman and Roberts, by showing that any partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for some $s$ such that $s < \frac{9}{4}k$ when $k$ is odd and $s < (6-2\sqrt{2})k$ when $k$ is even. For general $k$, these constants cannot be improved. We also obtain stronger results subject to placing a lower bound on $n$.

Information systems engineering with Digital Shadows: Concept and use cases in the Internet of Production

Entering the second decade of the Industrie 4.0 vision, the production sector is facing challenges in taking full advantage of global digitalization. Production research has focused on sophisticated mathematical models ranging from molecular materials modeling to production control to supply chain logistics. These models help simulate and control the related physical system but the variety of individual situations and behaviors is captured only as statistical uncertainty. The emergence of data-driven methods adds statistical or AI models learned from real-time production data to Digital Twins, and ideally allows for continuous synchronization (twinning) between physical and virtual system.However, the complexity of today’s production systems precludes Digital Twins covering more than just a few system perspectives, especially if realtime performance is required. To achieve better performance and more precise context adaptation, the interdisciplinary research cluster “Internet of Production” at RWTH Aachen University is exploring the concept of Digital Shadows. We conceptualize Digital Shadows as a generalization of compact views on dynamic processes, whose defining “query” combines condensed measurement data with efficient simplified mathematical models. Their small size makes Digital Shadows amenable to dynamic function allocation in hybrid cloud–edge settings. In addition to showing the similarities and differences to the traditional view concept, we also present a conceptual embedding of Digital Shadows in the context of large distributed system architectures, and sovereign data exchange in international Data Space communities. Two production use case experiences demonstrate that Digital Shadows can be valuable carriers of deep and reusable engineering knowledge for technical and ecological progress.

Local Multiset Dimension of Amalgamation Graphs

\textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs\\

Robust domination in random graphs

In this paper, we study “robust” dominating sets of random graphs that retain the domination property even if a small deterministic set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.

The expected values of the total numbers of independent edge sets and independent sets in random alpha-type pentagonal chains

A independent edge set of G containing mutually independent edges is also called a matching of G. The total numbers of matchings and independent sets of a graph G, namely, the Hosoya index and the Merrifield-Simmons index, respectively, are two important topological indices. We compute the average total numbers of independent edge sets and independent sets in random alpha-type pentagonal chains.

The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs

In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence. First we analyze the matching process, which builds a set of disjoint edges one edge at a time; then we analyze the independent process, which builds an independent set of vertices one vertex at a time. We use the differential equations method and apply a general theorem of Warnke. Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the matching or independent set.

Locally Balanced G-Designs

Let G be a graph and let Kn be the complete graph of order n. A G-design is a decomposition of the set of edges of Kn in graphs isomorphic to G, which are called blocks. It is well-known that a G-design is balanced if all the vertices are contained in the number of blocks of G. In this paper, the definition of locally balanced G-design is given, generalizing the existing concepts related to balanced designs. Further, locally balanced G-designs are studied in the cases in which G≅C4+e and G≅C4+P3, determining the spectrum.

Clustering constrained on linear networks

An unsupervised classification method for point events occurring on a geometric network is proposed. The idea relies on the distributional flexibility and practicality of random partition models to discover the clustering structure featuring observations from a particular phenomenon taking place on a given set of edges. By incorporating the spatial effect in the random partition distribution, induced by a Dirichlet process, one is able to control the distance between edges and events, thus leading to an appealing clustering method. A Gibbs sampler algorithm is proposed and evaluated with a sensitivity analysis. The proposal is motivated and illustrated by the analysis of crime and violence patterns in Mexico City.

Perfectly antimagic total graphs

An one-one correspondence function λ from V(G) ∪ E(G) to the set {1, 2, …, |V(G) | + |E(G) |} is a total labeling of a finite undirected graph G without loops and multiple edges, where |V(G) |and |E(G) | are the cardinality of vertex and edge set of G respectively. A perfectly antimagic total labeling is a totally antimagic total labeling whose vertex and edge-weights that are also pairwise distint. Perfectly antimagic total (PAT) graph is a graph having such labeling. The topic of discovering perfectly antimagic total labeling of some families of graphs is discussed in this paper. We also came up with certain conclusions about dual of a perfectly antimagic total graphs. Finally, we provided that the necessary and sufficient condition for a dual of a regular and irregular PAT graph to be a PAT graph.

Multistage s–t Path: Confronting Similarity with Dissimilarity

Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 of LNCS. Springer, pp 563–575, 2014), we provide a first, comprehensive study of finding a short s–t path in the multistage graph model, referred to as the Multistages–tPath problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s–t paths in each graph (“snapshot”) such that in the found path sequence the consecutive s–t paths are “similar”. We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistages–tPath are already {text {NP}}-hard for an input sequence of only two snapshots and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness ({text {W[1]}}-hardness) regarding the parameter path length (solution size). As a further conceptual investigation, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing so-called representative sets known from non-temporal settings), we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting with our W[1]-hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).

Evolutionary generation of test suites for multi-path coverage of MPI programs with non-determinism

When a large number of target paths in a sequential program need to be covered, we can divide similar target paths into the same group, and generate a test suite covering the same group of target paths at the same time, so as to reduce the testing cost. However, different communication edges may be run under a same test input when executing a Message-Passing <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</b> nterface (MPI) program with non-determinism, which cause different code fragments may be traversed, indicating the difficulty of generating a test suite to cover each group of target paths. This paper proposes an approach to evolutionary generation of test suites for multi-path coverage of MPI programs with non-determinism, which can significantly reduce the testing cost and difficulty. We first design an indicator for evaluating each traversal set of communication edges, which is used to form a relation matrix between each target path and each traversal set of communication edges, so as to divide all the target paths into a certain amount of groups. Then, we construct an optimization model for test suite generation associated with each group. Finally, an evolutionary optimization algorithm is extended to solve each model, and used to generate a test suite covering each group of target paths. The proposed approach is utilized and compared with several state-of-the-art approaches to seven benchmark MPI programs, as well as the experimental results illustrate that the proposed approach can efficiently generate a test suite, thus supporting the superiority of the proposed approach.

Road Topology Extraction From Satellite Imagery by Joint Learning of Nodes and Their Connectivity

Road topology extraction from satellite images, which has long been of interest, is an essential task in remote sensing. The graph representation of road networks is one of the most challenging aspects of road topology extraction. Most existing approaches cast road extraction as binary segmentation and then use post-processing, such as skeletonization, to infer networks from pixel-wise prediction. In our work, we believe that a road network can be represented by an undirected graph denoted as G=(V,E), where V and E represent the set of road nodes and the set of edges between nodes, respectively. Thus, to construct the road topology, we propose NodeConnect, a new method of extracting nodes for a road network and inferring the connectivity between nodes. A convolutional neural network is jointly trained to predict the nodes and connectivity map for nodes, and the edges between nodes are inferred from the connectivity map. We compare our approach with several segmentation methods on the DeepGlobe and RoadTracer datasets. The experiments show that our approach achieves state-of-the-art performance in terms of pixel-based metrics and topological precision and recall.

Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1

&lt;abstract&gt;&lt;p&gt;Let $ G = (V, E) $ be a simple graph with vertex set $ V $ and edge set $ E $, and let $ f $ be a function $ f:V\mapsto \{0, 1, 2\} $. A vertex $ u $ with $ f(u) = 0 $ is said to be undefended with respect to $ f $ if it is not adjacent to a vertex with positive weight. The function $ f $ is a weak Roman dominating function (WRDF) if each vertex $ u $ with $ f(u) = 0 $ is adjacent to a vertex $ v $ with $ f(v) &amp;gt; 0 $ such that the function $ f_{u}:V\mapsto \{0, 1, 2\} $, defined by $ f_{u}(u) = 1 $, $ f_{u}(v) = f(v)-1 $ and $ f_{u}(w) = f(w) $ if $ w\in V-\{u, v\} $, has no undefended vertex. The weight of $ f $ is $ w(f) = \sum_{v\in V}f(v) $. The weak Roman domination number, denoted $ \gamma_{r}(G) $, is the minimum weight of a WRDF in G. The domination number, denoted $ \gamma(G) $, is the minimum cardinality of a dominating set in $ G $. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 ($ \gamma_{r}(T) = \gamma(T)+1 $) by recursion and construction.&lt;/p&gt;&lt;/abstract&gt;

On the fractional total domatic numbers of incidence graphs

&lt;abstract&gt;&lt;p&gt;For a hypergraph $ H $ with vertex set $ X $ and edge set $ Y $, the incidence graph of hypergraph $ H $ is a bipartite graph $ I(H) = (X, Y, E) $, where $ xy\in E $ if and only if $ x\in X $, $ y\in Y $ and $ x\in y $. A total dominating set of graph $ G $ is a vertex subset that intersects every open neighborhood of $ G $. Let $ \mathscr{M} $ be a family of (not necessarily distinct) total dominating sets of $ G $ and $ r_{\mathscr{M}} $ be the maximum times that any vertex of $ G $ appears in $ \mathscr{M} $. The fractional domatic number $ G $ is defined as $ FTD(G) = \sup_{\mathscr{M}}\frac{|\mathscr{M}|}{r_{\mathscr{M}}} $. In 2018, Goddard and Henning showed that the incidence graph of every complete $ k $-uniform hypergraph $ H $ with order $ n $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq 2k\geq 4 $. We extend the result to the range $ n &amp;gt; k\geq 2 $. More generally, we prove that every balanced $ n $-partite complete $ k $-uniform hypergraph $ H $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq k $ and $ H\ncong K_n^{(n)} $, where $ FTD(I(K_n^{(n)})) = 1 $.&lt;/p&gt;&lt;/abstract&gt;

The existence of bipartite almost self-complementary 3-uniform hypergraphs

An almost self-complementary 3-uniform hypergraph on \(n\) vertices exists if and only if \(n\) is congruent to 3 modulo 4 A hypergraph \(H\) with vertex set \(V\) and edge set \(E\) is called bipartite if \(V\) can be partitioned into two subsets \(V_1\) and \(V_2\) such that \(e\cap V_1\neq \emptyset\) and \(e\cap V_2\neq \emptyset\) for any \(e\in E\). A bipartite self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of the vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) exists if and only if either (i) \(m=n\) or (ii) \(m\neq n\) and either \(m\) or \(n\) is congruent to 0 modulo 4 or (iii) \(m\neq n\) and both \(m\) and \(n\) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of a vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) and find the conditions on \(m\) and \(n\) for a bipartite 3-uniform hypergraph \(H\) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.

Sum index and difference index of graphs

Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to\mathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)\to\mathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E(G)\to\mathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.

Cayley bipolar fuzzy graphs associated with bipolar fuzzy groups

Recently, bipolar fuzzy graph is a growing research topic as it is the generalisation of fuzzy graphs. Let G be a non-trivial group and S be a non-empty subset of G such that not containing the identity element of G and S = S-1 = {s-1| s ∈ S}. The Cayley graph Γ = Cay (G, S) is the graph whose vertex set V(Γ) is G and edge set E(Γ) is {{g, gs}| g ∈ G, s ∈ S}. A non-empty subset S of G such that not containing the identity and S = S-1 is referred to as a Cayley subset of G, and the Cayley graph Γ = Cay (G, S) is referred to as a Cayley graph of G relative to S. In this paper, we introduce the concept of Cayley bipolar fuzzy graphs on the bipolar fuzzy groups. Also some properties of Cayley bipolar fuzzy graphs as connectivity and transitivity are provided.

Some new results on sum index and difference index

&lt;abstract&gt;&lt;p&gt;Let $ G = (V(G), E(G)) $ be a graph with a vertex set $ V(G) $ and an edge set $ E(G) $. For every injective vertex labeling $ f:V\left (G \right)\to \mathbb{Z} $, there are two induced edge labelings denoted by $ f^{+} :E\left (G \right)\to \mathbb{Z} $ and $ f^{-} :E\left (G \right)\to \mathbb{Z} $. These two edge labelings $ f^{+} $ and $ f^{-} $ are defined by $ f^{+}(uv) = f(u)+f(v) $ and $ f^{-}(uv) = \left |f(u)-f(v)\right | $ for each $ uv\in E(G) $ with $ u, v\in V(G) $. The sum index and difference index of $ G $ are induced by the minimum ranges of $ f^{+} $ and $ f^{-} $, respectively. In this paper, we obtain the properties of sum and difference index labelings. We also improve the bounds on the sum indices and difference indices of regular graphs and induced subgraphs of graphs. Further, we determine the sum and difference indices of various families of graphs such as the necklace graphs and the complements of matchings, cycles and paths. Finally, we propose some conjectures and questions by comparison.&lt;/p&gt;&lt;/abstract&gt;

On the maximum sigma index of k-cyclic graphs

Let G be a graph with edge set E ( G ) . Denote by d w the degree of a vertex w of G . The sigma index of G is defined as ∑ u v ∈ E ( G ) ( d u − d v ) 2 . A connected graph of order n and size n + k − 1 is known as a connected k -cyclic graph. Abdo et al. (2018) characterized the graphs having the greatest sigma index over the family of all connected graphs of a fixed order. The primary goal of the present note is to determine the unique graph possessing the greatest sigma index from the class of all connected k -cyclic graphs of order n for every pair of fixed integers k and n satisfying 0 ≤ k ≤ n − 2 .