# THE SEMI-SPLITTING BLOCK GRAPHS WITH CROSSING NUMBERS THREE AND FORBIDDEN SUBGRAPHS FOR CROSSING NUMBER ONE

## Main Article Content

## Abstract

Let *G = (V ,E)* be a simple connected undirected graph with vertex set *V* and edge set *E*. The advent of graph theory has played a prominent role in wide variety of engineering applications. Minimization of crossing numbers in graphs optimizes its use in many applications. In this paper, we establish a necessary and sufficient condition for semi-splitting block graph to have crossing number 3.We deduce a necessary and sufficient condition for semi-splitting block graph to have crossing number 1 in terms of forbidden subgraphs.

Keywords:

Semi-splitting, block, crossing number, forbidden, planar, minimally nonouterplanar

## Article Details

How to Cite

*Asian Journal of Current Research*,

*5*(1), 25-32. Retrieved from https://ikprress.org/index.php/AJOCR/article/view/5288

Issue

Section

Original Research Article

## References

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Kulli VR, Biradar MS. The middle blict graph of a graph. International Research Journal of Pure Algebra. 2015;5(7):111-117.

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Kulli VR, Biradar MS. The point block graphs and crossing numbers. Acta Ciencia Indica. 2007;33(2):637-640.

Kulli VR, Biradar MS. The line splitting graph of a graph. Acta Ciencia Indica. 2002;XXVIII(3):435.

Kulli VR, Niranjan KM. On minimally nonouterplanarity of the semi-total (point) graph of a graph. J. Sci. Res. 2009;1(3):551-557.

Kulli VR, Niranjan KM. The semi-image neighbourhood block graph of a graph. Asian Journal of Mathematics and Computer Research. 2020;27(2):36-41.

Kulli VR, Niranjan KM. The total closed neighbourhood graphs with crossing number three and four. Journal of Analysis and Computation. 2005;1(1):47-56.

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Niranjan KM. Forbidden subgraphs for planar and outerplanar interms of blict graphs. Journal of Analysis and Computation. 2006;2(1):19- 22.

Niranjan KM, Nagaraja P, Lokesh V. Semi-image neighborhood block graphs with crossing numbers. Journal of Scientific Research. 2013;5(2):295-299.

Niranjan KM, Poral Nagaraja, Lokesh V. Forbidden subgraphs for graphs with quasi-total graphs of crossing number ≤ 2. Journal of Intelligence System Research. 2008;2(2):109-113.

Rajendra Prasad KC, Niranjan KM, Venkanagouda M. Goudar. Vertex semi-middle graph of a graph. Malaya Journal of Matematik. 2019;7(4):786-789.

Rajendra Prasad KC, Venkanagouda M. Goudar, Niranjan KM. Pathos vertex semi-middle graph of a tree. South East Asian J. of Mathematics and Mathematical Sciences. 2020;16(1):171-176.

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Kulli VR, Niranjan KM. On minimally nonouterplanarity of a semi-splitting block graph of a graph. International Journal of Mathematics Trends and Technology. 2020;66(7):127-133.

Kulli VR, Niranjan KM. The semi-splitting block graphs with crossing numbers. Asian Journal of Current Research. 2020;5(1):9-16.

Kulli VR, Niranjan KM. The semi-splitting block graph of a graph. Journal of Scientific Research. 2010;2(3):485-488.

Biradar MS. Eulerianity of some graph valued functions. International Journal of Mathematics Trends and Technology. 2016;33(2):127-129.

Biradar MS, Kulli VR. Results on labeled path and its iterated line graphs. Intern. J. Fuzzy Mathematical Archive. 2016;10(2):125-129.

Biradar MS, Hiremath SS. The total blitact graph of a graph. Intern. J. Mathematical Archive. 2016;7(5):49-54.

Kulli VR. On minimally nonouterplanar graphs. Proc. Indian. Nat. Sci. Acad. 1975;41:275-280.

Kulli VR. The semitotal block graph and total-block graph of a graph. Indian J. Pure Appl. Math. 1976;7:625-630.

Kulli VR. On full graphs. J. Comp. & Math. Sci. 2015;6:261-267.

Kulli VR. On the plick graph and the qlick graph of a graph. Research Journal. 1988;1:48-52.

Kulli VR, Akka DG. Traversability and planarity of semitotal block graphs. J. Math. and Phy. Sci. 1978;12:177-178.

Kulli VR, Akka DG. Traversability and planarity of total block graphs. J. Mathematical and Physical Sciences. 1977;23(12):365-375.

Kulli VR, Akka DG. On semientire graphs. J. Math. and Phy. Sci. 1981;15:585-589.

Kulli VR, Akka DG. Characterization of minimally nonouterplanar graphs. J. Karnatak Univ. Sci. 1977;22:67-73.

Kulli VR, Annigeri NS. The ctree and total ctree of a graph. Vijnana Ganga. 1981;2:10-24.

Kulli VR, Basavanagoud B. On the quasivertex total graph of a graph. J. Karnatak University Sci. 1998;42:1-7.

Kulli VR, Basavanagoud B, Niranjan KM. Quasi-total graphs with crossing numbers. Journal of Scientific Research. 2010;2(2):257-263.

Kulli VR, Biradar MS. On eulerian blict graphs and blitact graphs. Journal of Computer and Mathematical Sciences. 2015;6(12):712-717.

Kulli VR, Biradar MS. The point block graph of a graph. Journal of Computer and Mathematical Sciences. 2014;5(5):476-481.

Kulli VR, Biradar MS. The middle blict graph of a graph. International Research Journal of Pure Algebra. 2015;5(7):111-117.

Kulli VR, Biradar MS. Planarity of the point block graph of a graph. Ultra Scientist. 2006;18:609-611.

Kulli VR, Biradar MS. The point block graphs and crossing numbers. Acta Ciencia Indica. 2007;33(2):637-640.

Kulli VR, Biradar MS. The line splitting graph of a graph. Acta Ciencia Indica. 2002;XXVIII(3):435.

Kulli VR, Niranjan KM. On minimally nonouterplanarity of the semi-total (point) graph of a graph. J. Sci. Res. 2009;1(3):551-557.

Kulli VR, Niranjan KM. The semi-image neighbourhood block graph of a graph. Asian Journal of Mathematics and Computer Research. 2020;27(2):36-41.

Kulli VR, Niranjan KM. The total closed neighbourhood graphs with crossing number three and four. Journal of Analysis and Computation. 2005;1(1):47-56.

Kulli VR, Warad NS. On the total closed neighbourhood graph of a graph. J. Discrete Mathematical Sciences and Cryptography. 2001;4:109-114.

Muddebihal MH, Usha P, Milind SC. Image neighbourhood graph of graph. The Mathematics Education. 2002;XXXVI(2).

Niranjan KM. Forbidden subgraphs for planar and outerplanar interms of blict graphs. Journal of Analysis and Computation. 2006;2(1):19- 22.

Niranjan KM, Nagaraja P, Lokesh V. Semi-image neighborhood block graphs with crossing numbers. Journal of Scientific Research. 2013;5(2):295-299.

Niranjan KM, Poral Nagaraja, Lokesh V. Forbidden subgraphs for graphs with quasi-total graphs of crossing number ≤ 2. Journal of Intelligence System Research. 2008;2(2):109-113.

Rajendra Prasad KC, Niranjan KM, Venkanagouda M. Goudar. Vertex semi-middle graph of a graph. Malaya Journal of Matematik. 2019;7(4):786-789.

Rajendra Prasad KC, Venkanagouda M. Goudar, Niranjan KM. Pathos vertex semi-middle graph of a tree. South East Asian J. of Mathematics and Mathematical Sciences. 2020;16(1):171-176.

Sampathkumar, Walikar HB. J. Karnatak Univ. Sci. 1980-81;25:13.

Kulli VR, Niranjan KM. On minimally nonouterplanarity of a semi-splitting block graph of a graph. International Journal of Mathematics Trends and Technology. 2020;66(7):127-133.

Kulli VR, Niranjan KM. The semi-splitting block graphs with crossing numbers. Asian Journal of Current Research. 2020;5(1):9-16.