A Criterion for (Non-)Planarity of The Block-Transformation Graph Gαβγ when αβγ = 101

Basavanagoud, B.; Veeragoudar, Jaishri B.

doi:10.18052/www.scipress.com/BMSA.10.38

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A Criterion for (Non-)Planarity of The Block-Transformation Graph G^αβγ when αβγ = 101

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Abstract:

The general concept of the block-transformation graph G^αβγ was introduced in [1]. The vertices and blocks of a graph are its members. The block-transformation graph G101 of a graph G is the graph, whose vertex set is the union of vertices and blocks of G, in which two vertices are adjacent whenever the corresponding vertices of G are adjacent or the corresponding blocks of G are nonadjacent or the corresponding members of G are incident. In this paper, we present characterizations of graphs whose block-transformation graphs G¹⁰¹ are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block-transformation graph G¹⁰¹ to have crossing number one.

Info:

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 10)

Pages:

38-47

DOI:

https://doi.org/10.18052/www.scipress.com/BMSA.10.38

Citation:

B. Basavanagoud and J. B. Veeragoudar, "A Criterion for (Non-)Planarity of The Block-Transformation Graph G^αβγ when αβγ = 101", Bulletin of Mathematical Sciences and Applications, Vol. 10, pp. 38-47, 2014

Online since:

November 2014

Authors:

B. Basavanagoud, Jaishri B. Veeragoudar

Keywords:

Crossing Number, Minimally Nonouterplanar, Outerplanar, Planar

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Open Access

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Creative Commons Attribution 4.0 International License

References:

B. Basavanagoud, H. P. Patil and Jaishri B. Veeragoudar, On the block-transformation graphs, graph-equations and diameters, International Journal of Advances in Science and Technology, Vol. 2, No. 2, (2011), 62-74.

G. Chartrand and F. Harary, Planar permutaion graphs, Ann. Inst. Henri Poincare Sec. B3(1967), pp.433-438.

R. K. Guy, Latest results on crossing numbers in recent trends in graph theory, Springer, New York, 1971, pp.143-156.

F. Harary, Graph theory, Addison-Wesley, Reading, Mass, (1969).

V. R. Kulli, On minimally nonouterplanar graphs, Proc. Indian Nat. Sci. Acad., Vol. 41A(1975), pp.275-280.

V. R. Kulli, The semitotal-block graph and the total-block graph of a graph, Indian J. Pure. Appl. Math, 7(1976), 625-630.

V. R. Kulli and H. P. Patil, Minimally nonouterplanar graphs and some graph valued functions, Karnatak Univ. Sci. J., Vol. 21(1976), pp.123-129.

V. R. Kulli and M. H. Muddebihal, Total-block graphs and semitotal-block graphs with crossing numbers, Far East J. Appl. Math. 4(1)(2000), 99-106.

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Cited By:

[1] A. Kelarev, W. Susilo, M. Miller, J. Ryan, "Ideals of Largest Weight in Constructions Based on Directed Graphs", Bulletin of Mathematical Sciences and Applications, Vol. 15, p. 8, 2016

DOI: https://doi.org/10.18052/www.scipress.com/BMSA.15.8

[2] B. Basavanagoud, V. Kulli, "Qlick Graphs with Crossing Number One", Bulletin of Mathematical Sciences and Applications, Vol. 17, p. 75, 2016

DOI: https://doi.org/10.18052/www.scipress.com/BMSA.17.75