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A GRASP for graph planarization
Networks, 1997Summary: A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. We describe a GRASP for the graph planarization problem, extending the heuristic of \textit{O. Goldschmidt} and \textit{A. Takvorian} [Networks 24, No. 2, 69-73 (1994; Zbl 0789.90083)].
Mauricio G. C. Resende, Celso C. Ribeiro
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Generalizations of planar graphs
Networks, 1982AbstractTwo new generalizations of planar graphs, called quasiplanar and pseudoplanar graphs, are introduced and discussed. It is shown that planar graphs are quasiplanar and these in turn are pseudoplanar. Conversely, a pseudoplanar graph that contains with each arc its reverse arc is quasiplanar.
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Planarity for clustered graphs
1995In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study C-planarity of clustered graphs.
Qing-Wen Feng +2 more
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Clique graphs of planar graphs.
Ars Comb., 2004The main result of this paper is a characterization of those \(K_3\)-free or \(K_4\)-free graphs which occur as the clique graphs of planar graphs. Several examples are given of planar graphs which do not occur as clique graphs of planar graphs.
Liliana Alcón, Marisa Gutierrez
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Journal of Graph Theory, 1997
Let \(G\) be a 2-connected plane graph and let \(X_G\) be the circuit bounding the infinite face. This paper generalizes the following result by \textit{C. Thomassen} [J. Graph Theory 7, 169-176 (1983; Zbl 0515.05040)], which in turn was an improvement of an earlier theorem by \textit{W. T. Tutte} [Trans. Am. Math. Soc. 82, 99-116 (1956; Zbl 0070.18403)
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Let \(G\) be a 2-connected plane graph and let \(X_G\) be the circuit bounding the infinite face. This paper generalizes the following result by \textit{C. Thomassen} [J. Graph Theory 7, 169-176 (1983; Zbl 0515.05040)], which in turn was an improvement of an earlier theorem by \textit{W. T. Tutte} [Trans. Am. Math. Soc. 82, 99-116 (1956; Zbl 0070.18403)
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Canadian Mathematical Bulletin, 1968
In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs.
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In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs.
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Australas. J Comb., 1998
It is known that for an arbitrary graph on \(n\) vertices, the minimum time required to broadcast is \(\lceil \log_2n\rceil\), and for any \(n\), there exist graphs on \(n\) vertices with broadcast time equal to \(\lceil\log_2 n\rceil\). When restricted to planar graphs, this is generally not the case. In this paper, the planar broadcast time for \(n\),
Pavol Hell, Karen Seyffarth
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It is known that for an arbitrary graph on \(n\) vertices, the minimum time required to broadcast is \(\lceil \log_2n\rceil\), and for any \(n\), there exist graphs on \(n\) vertices with broadcast time equal to \(\lceil\log_2 n\rceil\). When restricted to planar graphs, this is generally not the case. In this paper, the planar broadcast time for \(n\),
Pavol Hell, Karen Seyffarth
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Interval valued m-polar fuzzy planar graph and its application
Artificial Intelligence Review, 2020Ganesh Ghorai +2 more
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The maximum number of paths of length four in a planar graph
Discrete Mathematics, 2021Ervin Gyori +2 more
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Every planar graph with girth at least 5 is (1,9)-colorable
Discrete Mathematics, 2022Xiangwen Li
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