Analytical modeling on the coloring of certain graphs for applications of air traffic and air scheduling management

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Manjula T. ◽  
Rajeswari R. ◽  
Praveenkumar T.R.
Keyword(s):  
Graph Coloring ◽  
Line Graph ◽  
Domination Number ◽  
Air Traffic ◽  
Airline Scheduling ◽  
Total Graph ◽  
Content Type ◽  
Coloring Number ◽  
Color Class ◽  
Dominator Coloring

Purpose The purpose of this paper is to assess the application of graph coloring and domination to solve the airline-scheduling problem. Graph coloring and domination in graphs have plenty of applications in computer, communication, biological, social, air traffic flow network and airline scheduling. Design/methodology/approach The process of merging the concept of graph node coloring and domination is called the dominator coloring or the χ_d coloring of a graph, which is defined as a proper coloring of nodes in which each node of the graph dominates all nodes of at least one-color class. Findings The smallest number of colors used in dominator coloring of a graph is called the dominator coloring number of the graph. The dominator coloring of line graph, central graph, middle graph and total graph of some generalized Petersen graph P_(n ,1) is obtained and the relation between them is established. Originality/value The dominator coloring number of certain graph is obtained and the association between the dominator coloring number and domination number of it is established in this paper.

scholarly journals Split and Non-Split Dominator Chromatic Numbers and Related Parameters

2011 ◽  
Vol 10 (1) ◽  
pp. 52-62
Author(s):  
K Kavitha ◽  
N G David ◽  
N. Selvi
Keyword(s):  
Graph Coloring ◽  
Proper Coloring ◽  
Chromatic Numbers ◽  
Color Class ◽  
Minimum Number ◽  
Dominator Coloring ◽  
Cycle Path

A proper graph coloring is defined as coloring the nodes of a graph with the minimum number of colors without any two adjacent nodes having the same color.  Dominator coloring of G is a proper coloring in which every vertex of G dominates every vertex of at least one color class.  In this paper, new parameters, namely strong split and non-split dominator chromatic numbers and block, cycle, path non-split dominator chromatic numbers are introduced.  These parameters are obtained for different classes of graphs and also interesting results are established.

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

Understanding aviation operators’ variability in advanced systems

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Maria Papanikou ◽  
Utku Kale ◽  
András Nagy ◽  
Konstantinos Stamoulis
Keyword(s):  
Human Performance ◽  
Situational Awareness ◽  
Air Traffic ◽  
Content Type ◽  
Homogeneous Groups ◽  
Professional Groups ◽  
Performance Shaping Factors ◽  
Generational Changes ◽  
And Performance ◽  
Practical Implications

Purpose This study aims to identify variability in aviation operators in order to gain greater understanding of the changes in aviation professional groups. Research has commonly addressed human factors and automation in broad categories according to a group’s function (e.g., pilots, air traffic controllers [ATCOs], engineers). Accordingly, pilots and Air Traffic Controls (ATCOs) have been treated as homogeneous groups with a set of characteristics. Currently, critical themes of human performance in light of systems’ developments place the emphasis on quality training for improved situational awareness (SA), decision-making and cognitive load. Design/methodology/approach As key solutions centre on the increased understanding and preparedness of operators through quality training, the authors deploy an iterative mixed methodology to reveal generational changes of pilots and ATCOs. In total, 46 participants were included in the qualitative instrument and 70 in the quantitative one. Preceding their triangulation, the qualitative data were analysed using NVivo and the quantitative analysis was aided through descriptive statistics. Findings The results show that there is a generational gap between old and new generations of operators. Although positive views on advanced systems are being expressed, concerns about cognitive capabilities in the new systems, training and skills gaps, workload and role implications are presented. Practical implications The practical implications of this study extend to different profiles of operators that collaborate either directly or indirectly and that are critical to aviation safety. Specific implications are targeted on automation complacency, bias and managing information load, and training aspects where quality training can be aided by better understanding the occupational transitions under advanced systems. Originality/value In this paper, the authors aimed to understand the changing nature of the operators’ profession within the advanced technological context, and the perceptions and performance-shaping factors of pilots and ATCOs to define the generational changes.

scholarly journals ON DYNAMIC COLORING OF WEB GRAPH

2018 ◽  
Vol 5 (2) ◽  
pp. 11-15
Author(s):  
Aaresh R.R ◽  
Venkatachalam M ◽  
Deepa T
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Web Graph ◽  
Middle Graph

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals Omega Index of Line and Total Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Musa Demirci ◽  
Sadik Delen ◽  
Ahmet Sinan Cevik ◽  
Ismail Naci Cangul
Keyword(s):  
Line Graph ◽  
Graph Invariant ◽  
Original Graph ◽  
Total Graph ◽  
Graph Classes ◽  
Number Of Faces

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

scholarly journals Equitable power domination number of total graph of certain graphs

2020 ◽  
Vol 1531 ◽  
pp. 012073
Author(s):  
S Banu Priya ◽  
A Parthiban ◽  
P Abirami
Keyword(s):  
Domination Number ◽  
Total Graph ◽  
Power Domination

scholarly journals Hyperbolicity on Graph Operators

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 360 ◽  
Author(s):  
J. Méndez-Bermúdez ◽  
Rosalío Reyes ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Line Graph ◽  
Discrete Mathematics ◽  
Total Graph ◽  
Subdivision Graph ◽  
Gromov Product

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

scholarly journals Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 366 ◽  
Author(s):  
Jia-Bao Liu ◽  
Bahadur Ali ◽  
Muhammad Aslam Malik ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Muhammad Imran
Keyword(s):  
Physical Properties ◽  
Chemical Structure ◽  
Topological Index ◽  
Chemical Reactivity ◽  
Line Graph ◽  
Total Graph ◽  
Zagreb Indices ◽  
Subdivision Graph ◽  
Chemical Constitution ◽  
Vertex Degrees

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

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