In this article we admit Bi- Conditional for Duplicate graph of Quadrilateral Snake EDG(QSm)m ≥ 2, Double Quadrilateral SnakeGraphs EDG(DQSm)m ≥ 2, and Triangular ladder graph. EDG(TQLm)m ≥ 2.
Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically ...draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing nonoverlapping station labels. In this paper, we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NP-hard) using mixed-integer programming (MIP). We identify seven design rules used in most real-world metro maps. We split these rules into hard and soft constraints and translate them into an MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods.
As discussed at length in Christodoulakis et al. (2015) 3, there is a natural one-many correspondence between simple undirected graphs G with vertex set V={1,2,…,n} and indeterminate stringsx=x1..n — ...that is, sequences of subsets of some alphabet Σ. In this paper, given G, we consider the “reverse engineering” problem of computing a corresponding x on an alphabet Σmin of minimum cardinality. This turns out to be equivalent to the NP-hard problem of computing the intersection number of G, thus in turn equivalent to the clique cover problem. We describe a heuristic algorithm that computes an approximation to Σmin and a corresponding x. We give various properties of our algorithm, including some experimental evidence that on average it requires O(n2logn) time. We compare it with other heuristics, and state some conjectures and open problems.
•Algorithm reverse engineering an indeterminate string from an arbitrary undirected graph.•Our algorithm is simple and faster than other algorithms, though it yields larger alphabets.•We clarify the connections between clique covers, intersection numbers and alphabet sizes for indeterminate strings.•We have created a software suite for running experiments, and we present several experimental results in the paper.
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.
Supermagic Graphs with Many Different Degrees Kovář, Petr; Kravčenko, Michal; Silber, Adam ...
Discussiones Mathematicae. Graph Theory,
11/2021, Volume:
41, Issue:
4
Journal Article
Peer reviewed
Open access
Let
= (
) be a graph with
vertices and
edges. A supermagic labeling of
is a bijection
from the set of edges
to a set of consecutive integers {
+ 1, . . . ,
+
− 1} such that for every vertex
∈
the sum ...of labels of all adjacent edges equals the same constant
. This
is called a magic constant of
, and
is a supermagic graph.
The existence of supermagic labeling for certain classes of graphs has been the scope of many papers. For a comprehensive overview see Gallian’s
in the Electronic Journal of Combinatorics. So far, regular or almost regular graphs have been studied. This is natural, since the same magic constant has to be achieved both at vertices of high degree as well as at vertices of low degree, while the labels are distinct consecutive integers.
The modern world has brought everyone closer through digital communications. People are highly dependent on digital services. The quest for a more complex coding algorithm to prevent data from ...breaching is never-ending. The more intricate the algorithm is, the safer the communication will be. So, the vogue is to find the most convoluted algorithm to provide secure communication. In this paper, a unique algorithm is developed using graph labeling, a complement of a graph and generalized complement. The algorithm generates two labeled graphs satisfying vertex even mean and vertex odd mean labelings. Encryption involves the process of complementation and combining both graphs by k-complementation of graphs to obtain a cipher graph. The reverse process is applied for decryption which involves splitting the cipher graph into two subgraphs by applying k-complement for the specified partition of the vertex set, taking the complement of the obtained graph/s and getting the values of the plaintext using the graph labeling method. The proposed algorithm is designed in such a way that it should be useful for all kinds of plaintexts even with special characters. To illustrate this, an app is developed in the Android platform for communication of messages using endto-end encryption.
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