Research

11 pages, 385 KiB

Open AccessArticle

Rainbow Connection Numbers of WK-Recursive Networks and WK-Recursive Pyramids

by Fu-Hsing Wang and Cheng-Ju Hsu

Mathematics 2024, 12(7), 944; https://doi.org/10.3390/math12070944 - 22 Mar 2024

Viewed by 381

An edge coloring of a graph G results in G being rainbow connected when every pair of vertices is linked by a rainbow path. Such a path is defined as one where each edge possesses a distinct color. A rainbow coloring refers to [...] Read more.

An edge coloring of a graph G results in G being rainbow connected when every pair of vertices is linked by a rainbow path. Such a path is defined as one where each edge possesses a distinct color. A rainbow coloring refers to an edge coloring that guarantees the rainbow connectedness of G. The rainbow connection number of G represents the smallest quantity of colors required to achieve rainbow connectedness under a rainbow coloring scheme. Wang and Hsu (ICICM 2019: 75–79) provided upper bounds on the size of the rainbow connection numbers in WK-recursive networks

{WK}_{d, t}

and WK-recursive pyramids

{WKP}_{d, n}

. In this paper, we revise their results and determine the exact values of the rainbow connection numbers of

{WK}_{d, 2}

for

d = 3

and 4. The rainbow connection numbers of

{WK}_{d, 2}

are bounded between 4 and

⌊ \frac{d}{2} ⌋ + 2

for

d > 4

. In addition to our previous findings, we further investigate and determine upper bounds for the size of the rainbow connection numbers of

{WKP}_{d, n}

. This involves analyzing various aspects of the graph structure and exploring potential limitations on the rainbow connection numbers. By establishing these upper bounds, we gain deeper insights into the potential range and constraints of the rainbow connection numbers within the given context. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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20 pages, 2383 KiB

Open AccessArticle

An Efficient GNSS Coordinate Classification Strategy with an Adaptive KNN Algorithm for Epidemic Management

by Jong-Shin Chen and Chun-Ming Kuo

Mathematics 2024, 12(4), 536; https://doi.org/10.3390/math12040536 - 08 Feb 2024

Viewed by 480

Abstract

In times of widespread epidemics, numerous individuals are at risk of contracting viruses, such as COVID-19, monkeypox, and pneumonia, leading to a ripple effect of impacts on others. Consequently, the Centers for Disease Control (CDC) typically devises strategies to manage the situation by [...] Read more.

In times of widespread epidemics, numerous individuals are at risk of contracting viruses, such as COVID-19, monkeypox, and pneumonia, leading to a ripple effect of impacts on others. Consequently, the Centers for Disease Control (CDC) typically devises strategies to manage the situation by monitoring and tracing the infected individuals and their areas. For convenience, “targets” and “areas” represent the following individuals and areas. A global navigation satellite system (GNSS) can assist in evaluating the located areas of the targets with pointing-in-polygon (PIP) related technology. When there are many targets and areas, relying solely on PIP technology for classification from targets to areas could be more efficient. The classification technique of k-nearest neighbors (KNN) classification is widely utilized across various domains, offering reliable classification accuracy. However, KNN classification requires a certain quantity of targets with areas (training dataset) for execution, and the size of the training dataset and classification time often exhibit an exponential relationship. This study presents a strategy for applying KNN technology to classify targets into areas. Additionally, within the strategy, we propose an adaptive KNN algorithm to enhance the efficiency of the classification procedure. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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18 pages, 517 KiB

Open AccessArticle

A Validation of the Phenomenon of Linearly Many Faults on Burnt Pancake Graphs with Its Applications

by Mei-Mei Gu, Hong-Xia Yan and Jou-Ming Chang

Mathematics 2024, 12(2), 268; https://doi.org/10.3390/math12020268 - 14 Jan 2024

Viewed by 509

Abstract

“Linearly many faults” is a phenomenon observed by Cheng and Lipták in which a specific structure emerges when a graph is disconnected and often occurs in various interconnection networks. This phenomenon means that if a certain number of vertices or edges are deleted [...] Read more.

“Linearly many faults” is a phenomenon observed by Cheng and Lipták in which a specific structure emerges when a graph is disconnected and often occurs in various interconnection networks. This phenomenon means that if a certain number of vertices or edges are deleted from a graph, the remaining part either stays connected or breaks into one large component along with smaller components with just a few vertices. This phenomenon can be observed in many types of graphs and has important implications for network analysis and optimization. In this paper, we first validate the phenomenon of linearly many faults for surviving graph of a burnt pancake graph

B P_{n}

when removing any edge subset with a size of approximately six times

λ (B P_{n})

. For graph G, the ℓ-component edge connectivity denoted as

λ_{ℓ} (G)

(resp., the ℓ-extra edge connectivity denoted as

λ^{(ℓ)} (G)

) is the cardinality of a minimum edge subset S such that

G - S

is disconnected and has at least ℓ components (resp., each component of

G - S

has at least

ℓ + 1

vertices). Both

λ_{ℓ} (G)

and e

λ^{(ℓ)} (G)

are essential metrics for network reliability assessment. Specifically, from the property of “linearly many faults”, we may further prove that

λ_{5} (B P_{n}) = λ^{(3)} (B P_{n}) + 3 = 4 n - 3

for

n ⩾ 5

;

λ_{6} (B P_{n}) = λ^{(4)} (B P_{n}) + 4 = 5 n - 4

and

λ_{7} (B P_{n}) = λ^{(5)} (B P_{n}) + 5 = 6 n - 5

for

n ⩾ 6

. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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12 pages, 468 KiB

Open AccessArticle

Two Combinatorial Algorithms for the Constrained Assignment Problem with Bounds and Penalties

by Guojun Hu, Junran Lichen and Pengxiang Pan

Mathematics 2023, 11(24), 4883; https://doi.org/10.3390/math11244883 - 06 Dec 2023

Viewed by 609

Abstract

In the paper, we consider a generalization of the classical assignment problem, which is called the constrained assignment problem with bounds and penalties (CA-BP). Specifically, given a set of machines and a set of independent jobs, each machine has a lower and upper [...] Read more.

In the paper, we consider a generalization of the classical assignment problem, which is called the constrained assignment problem with bounds and penalties (CA-BP). Specifically, given a set of machines and a set of independent jobs, each machine has a lower and upper bound on the number of jobs that can be executed, and each job must be either executed on some machine with a given processing time or rejected with a penalty that we must pay for. No job can be executed on more than one machine. We aim to find an assignment scheme for these jobs that satisfies the constraints mentioned above. The objective is to minimize the total processing time of executed jobs as well as the penalties from rejected jobs. The CA-BP is related to some practical applications such as edge computing, which involves selecting tasks and processing them on the edge servers of an internet network. As a result, a motivation of this study is to improve the efficiency of internet networks by limiting the lower bound of the number of objects processed by each edge server. Our main contribution is modifying the previous network flow algorithms to satisfy the lower capacity constraints, for which we design two exact combinatorial algorithms to solve the CA-BP. Our methodologies and results bring novel perspectives into other research areas related to the assignment problem. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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17 pages, 733 KiB

Open AccessArticle

Edge-Based Minimal k-Core Subgraph Search

by Ting Wang, Yu Jiang, Jianye Yang and Lei Xing

Mathematics 2023, 11(15), 3407; https://doi.org/10.3390/math11153407 - 04 Aug 2023

Viewed by 884

Abstract

In social networks, k-core is commonly used to measure the stability of a network. When a user in a k-core leaves the network, other users may follow the user to leave. Hence, maintaining a key user is important to keep the [...] Read more.

In social networks, k-core is commonly used to measure the stability of a network. When a user in a k-core leaves the network, other users may follow the user to leave. Hence, maintaining a key user is important to keep the stability of a network. It is known that an edge between two users models the relationship between the two users. In some scenarios, maintaining a relationship comes at a cost. Therefore, selectively in maintaining the relationships between users is crucial. In this paper, we for the first time conceive the concept of an edge-based minimal k-core model. An edge-based minimal k-core is a k-core with a minimal number of edges. In other words, removing any edge in an edge-based minimal k-core would make it not be a k-core any more. Based on this model, we proposed two problems, namely, an edge-based minimal k-core subgraph search (EMK-SS) and an edge-based minimal k-core subgraph search with a query node q (EMK-q-SS). Given a graph G, an integer k, and a query node (a key user) q, the EMK-q-SS problem is to find all the edge-based minimal k-cores containing the query node q, and the EMK-SS problem is to find all the edge-based minimal k-cores. We also theoretically prove that the two problems are both NP-complete. To deal with the proposed problems, we design two novel algorithms, namely the edge deletion algorithm and edge extension algorithm. Further, a graph partitioning technique is employed to speed up the computation. Comprehensive experiments on synthetic and real networks are conducted to demonstrate the effect and efficiency of our proposed methods. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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14 pages, 1278 KiB

Open AccessArticle

Cycle Existence for All Edges in Folded Hypercubes under Scope Faults

by Che-Nan Kuo and Yu-Huei Cheng

Mathematics 2023, 11(15), 3391; https://doi.org/10.3390/math11153391 - 03 Aug 2023

Viewed by 512

Abstract

The reliability of large-scale networks can be compromised by various factors such as natural disasters, human-induced incidents such as hacker attacks, bomb attacks, or even meteorite impacts, which can lead to failures in scope of processors or links. Therefore, ensuring fault tolerance in [...] Read more.

The reliability of large-scale networks can be compromised by various factors such as natural disasters, human-induced incidents such as hacker attacks, bomb attacks, or even meteorite impacts, which can lead to failures in scope of processors or links. Therefore, ensuring fault tolerance in the interconnection network is vital to maintaining system reliability. The n-dimensional folded hypercube network structure, denoted as

{F Q}_{n}

, is constructed by adding an edge between every pair of vertices with complementary addresses from an n-dimensional hypercube,

Q_{n}

. Notably,

{F Q}_{n}

exhibits distinct characteristics based on the dimensionality: it is bipartite for odd integers

n \geq 3

and non-bipartite for even integers

n \geq 2

. Recently, in terms of the issue of how

{F Q}_{n}

performs in communication under regional or widespread destruction, we mentioned that in

{F Q}_{n}

, even when a pair of adjacent vertices encounter errors, any fault-free edge can still be embedded in cycles of various lengths. Additionally, even when the smallest communication ring experiences errors, it is still possible to embed cycles of any length. The smallest communication ring in

{F Q}_{n}

is observed to be the four-cycle ring. In order to further investigate the communication capabilities of

{F Q}_{n}

, we further discuss whether every fault-free edge will still be a part of every communication ring with different lengths when the smallest communication ring is compromised in

{F Q}_{n}

. In this study, we consider a fault-free edge

e = (u, v)

and

F_{4} = {f_{1}, f_{2}, f_{3}, f_{4}}

as the set of faulty extreme vertices for any four cycles in

{F Q}_{n}

. Our research focuses on investigating the cycle-embedding properties in

{F Q}_{n} - F_{4}

, where the fault-free edges play a significant role. The following properties are demonstrated: (1) For

n \geq 4

in

{F Q}_{n} - F_{4}

, every even length cycle with a length ranging from

4 t o 2^{n} - 4

contains a fault-free edge e; (2) For every even

n \geq 4

in

{F Q}_{n} - F_{4}

, every odd length cycle with a length ranging from

n + 1 t o 2^{n} - 5

contains a fault-free edge

e

. These findings provide insights into the cycle-embedding capabilities of

{F Q}_{n}

, specifically in the context of fault tolerance when considering certain sets of faulty vertices. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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33 pages, 594 KiB

Open AccessArticle

Clique Transversal Variants on Graphs: A Parameterized-Complexity Perspective

by Chuan-Min Lee

Mathematics 2023, 11(15), 3325; https://doi.org/10.3390/math11153325 - 28 Jul 2023

Viewed by 648

Abstract

The clique transversal problem and its variants have garnered significant attention in the last two decades due to their practical applications in communication networks, social-network theory and transceiver placement for cellular telephones. While previous research primarily focused on determining the polynomial-time solvability or [...] Read more.

The clique transversal problem and its variants have garnered significant attention in the last two decades due to their practical applications in communication networks, social-network theory and transceiver placement for cellular telephones. While previous research primarily focused on determining the polynomial-time solvability or NP-hardness/NP-completeness of specific graphs, this paper adopts a parameterized-complexity approach. It thoroughly explores four clique transversal variants: the d-fold transversal problem, the

{d}

-clique transversal problem, the signed clique transversal problem and the minus clique transversal problem. The paper presents various findings regarding the parameterized complexity of the clique transversal problem and its variants. It establishes the W[2]-completeness and para-NP-completeness of the d-fold transversal problem, the

{d}

-clique transversal problem, the signed clique transversal problem and the minus clique transversal problem within specific graph classes. Additionally, it introduces fixed-parameter tractable algorithms for planar graphs and graphs with bounded treewidth, offering efficient solutions for these specific instances of the problems. The research further explores the relationship between planar graphs and graphs with bounded treewidth to enhance the time complexity of the d-fold clique transversal problem and the

{d}

-clique transversal problem. By analyzing the parameterized complexity of the clique transversal problem and its variants, this research contributes to our understanding of the computational limitations and potentially efficient algorithms for solving these problems. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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26 pages, 1096 KiB

Open AccessArticle

The Longest (s, t)-Path Problem on O-Shaped Supergrid Graphs

by Fatemeh Keshavarz-Kohjerdi and Ruo-Wei Hung

Mathematics 2023, 11(12), 2712; https://doi.org/10.3390/math11122712 - 15 Jun 2023

Cited by 1 | Viewed by 1560

Abstract

The longest

(s, t)

-path problem on supergrid graphs is known to be NP-complete. However, the complexity of this problem on supergrid graphs with or without holes is still unknown.In the past, we presented linear-time algorithms for solving the longest [...] Read more.

The longest

(s, t)

-path problem on supergrid graphs is known to be NP-complete. However, the complexity of this problem on supergrid graphs with or without holes is still unknown.In the past, we presented linear-time algorithms for solving the longest

(s, t)

-path problem on L-shaped and C-shaped supergrid graphs, which form subclasses of supergrid graphs without holes. In this paper, we will determine the complexity of the longest

(s, t)

-path problem on O-shaped supergrid graphs, which form a subclass of supergrid graphs with holes. These graphs are rectangular supergrid graphs with rectangular holes. It is worth noting that O-shaped supergrid graphs contain L-shaped and C-shaped supergrid graphs as subgraphs, but there is no inclusion relationship between them. We will propose a linear-time algorithm to solve the longest

(s, t)

-path problem on O-shaped supergrid graphs. The longest

(s, t)

-paths of O-shaped supergrid graphs have applications in calculating the minimum trace when printing hollow objects using computer embroidery machines and 3D printers. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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14 pages, 308 KiB

Open AccessArticle

An Algorithm for the Numbers of Homomorphisms from Paths to Rectangular Grid Graphs

by Hatairat Yingtaweesittikul, Sayan Panma and Penying Rochanakul

Mathematics 2023, 11(11), 2587; https://doi.org/10.3390/math11112587 - 05 Jun 2023

Viewed by 895

Abstract

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a

h o m o m o r p h i s m

from G to H if, for every pair [...] Read more.

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a

h o m o m o r p h i s m

from G to H if, for every pair of adjacent vertices x and y in G, the vertices

f (x)

and

f (y)

are adjacent in H. A rectangular grid graph is the Cartesian product of two path graphs. In this paper, we provide a formula to determine the number of homomorphisms from paths to rectangular grid graphs. This formula gives the solution to the problem concerning the number of walks in the rectangular grid graphs. Full article

(This article belongs to the Special Issue Graph Theory: Advanced Algorithms and Applications)

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