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Bipolar Fuzzy Planar Graphs

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Graphs for the Analysis of Bipolar Fuzzy Information

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 401))

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Abstract

In this chapter, we study the concepts of bipolar fuzzy multisets, bipolar fuzzy multigraphs, strong and complete bipolar fuzzy multigraphs, bipolar fuzzy planar graphs, and bipolar fuzzy dual graphs. We discuss different types of bipolar fuzzy edges, intersection value and planarity value of bipolar fuzzy graphs, strong and weak bipolar fuzzy faces, and the relation of planarity and duality in bipolar fuzzy graphs. We elaborate various properties of bipolar fuzzy bridges, bipolar fuzzy cut vertices, bipolar fuzzy blocks, bipolar fuzzy cycles, and bipolar fuzzy trees in terms of level graphs. We describe the importance of bipolar fuzzy planar graphs with a number of real-world applications in road networks and electrical connections. The main results of this chapter are from [6, 7].

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Lahore, Pakistan

    Muhammad Akram

  2. Department of Mathematics, Government College Women University, Sialkot, Pakistan

    Musavarah Sarwar

  3. Faculty of Pure and Applied Mathematics, Wrocław University of Technology, Wrocław, Poland

    Wieslaw A. Dudek

Authors
  1. Muhammad Akram
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  2. Musavarah Sarwar
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  3. Wieslaw A. Dudek
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Corresponding author

Correspondence to Muhammad Akram .

Exercises 5

Exercises 5

  1. 1.

    Let G and H be connected bipolar fuzzy fuzzy graphs different from \(K_1\) and \(K_2\), then prove or disprove that \(G\Box H\) is a bipolar fuzzy planar graph if and only if both factors are bipolar fuzzy paths, or one is a bipolar fuzzy path and other is a bipolar fuzzy cycle.

  2. 2.

    If \(G = (A, B)\) is a bipolar fuzzy graph with p vertices, then show that G has at most \(p - 1\) bipolar fuzzy bridges.

  3. 3.

    Prove that a nontrivial bipolar fuzzy tree H has at least two bipolar fuzzy end nodes.

  4. 4.

    If H is a bipolar fuzzy tree, then show that every vertex of H is either a bipolar fuzzy cut-node or a bipolar fuzzy end node. Also, show that the converse is not true.

  5. 5.

    If G is a bipolar fuzzy tree, then prove or disprove that the arcs of G are bipolar fuzzy bridges.

  6. 6.

    Show that a bipolar fuzzy cut vertex of a bipolar fuzzy tree is the common vertex of at least two bipolar fuzzy bridges.

  7. 7.

    If G is a bipolar fuzzy tree, then show that every vertex of G is either a bipolar fuzzy cut vertex or a bipolar fuzzy end node.

  8. 8.

    If G is a bipolar fuzzy tree, then show that it has at most \(n - 1\) number of bipolar fuzzy bridges where n is the number of vertices in G.

  9. 9.

    Show that every bipolar fuzzy bridge is strong, but a strong edge need not be a bipolar fuzzy bridge.

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Akram, M., Sarwar, M., Dudek, W.A. (2021). Bipolar Fuzzy Planar Graphs. In: Graphs for the Analysis of Bipolar Fuzzy Information. Studies in Fuzziness and Soft Computing, vol 401. Springer, Singapore. https://doi.org/10.1007/978-981-15-8756-6_5

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  • DOI: https://doi.org/10.1007/978-981-15-8756-6_5

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-15-8755-9

  • Online ISBN: 978-981-15-8756-6

  • eBook Packages: EngineeringEngineering (R0)

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