Volume-07 , Special Issue-05 , Mar - 2019 Go Back

Graph embedding is an important and extensively studied theory in parallel computing. A great deal of research has been devoted to finding “good” embedding of one network into another. The embedding of a guest graph G into a host graph H is defined by a pair of injective functions between them. The edge congestion of an embedding is the maximum number of edges of the guest graph that are embedded on any single edge e of the host graph. The optimal layout problem deals with finding the embedding for which the sum of all the shortest paths in H corresponding to the edges in G is minimum. In this paper, we find the optimal layout of embedding the chord graph into the windmill graph.

Embedding, chord graph, optimal set, windmill graph, layout

[1] J. Abraham, “Optimal Node Ordering and Layout of Chord Graphs”, International Journal of pure and Applied Mathematics, vol. 119, no. 11, pp. 19-27, 2018.
[2] M. Arockiaraj, J. Abraham, J. Quadras, A.J. Shalini, “Linear layout of locally twisted cubes”, International Journal of Computer Mathematics, vol. 94, no. 1, pp. 56-65, 2017.
[3] M. Arockiaraj, J. Quadras, I. Rajasingh, A.J. Shalini, “Embedding hypercubes and folded hypercubes onto Cartesian product of certain trees”, Discrete Optimization, vol. 17, pp. 1-13,2015.
[4] Garey, M.R. and Johnson, D.S. “Computers and Intractability,” A Guide to the Theory of NP- Completeness, Freeman, San Francisco.
[5] N. Parthiban, J. Ryan, I. Rajasingh, R.S. Rajan, L.N. Rani, “Exact Wirelength of embedding chord graph into tree-based architectures”, Int.J.Networking and Virtual Organisation , vol. 17, no. 1, pp. 76-87, 2017.
[6] I. Rajasingh, P. Manuel, M. Arockiaraj, B. Rajan, “Embedding of circulant networks”, Journal of Combinatorial. Optimization, vol. 26, no. 1, pp. 135-151, 2013.
[7] I. Rajasingh, B. Rajan, R.S. Rajan, “Embedding of hypercubes into necklace, Windmill and Snake graphs”, Information Processing Letter , vol. 112, pp. 509–515, 2012.
[8] Rajasingh, I., Manuel, P.,Rajan, B. and Arockiaraj, M. “Wirelength of hypercubes into certain trees”, Discrete Applied Mathematics, vol.160, No. 18, pp.2778-2786, 2012a.
[9] H. Rostami, J. Habibi, “Minimum linear arrangement of chord graphs”, Applied Mathematics and Computation, vol. 203, pp. 358–367, 2008.
[10] Xu, J.M. and Ma, M. “Survey on path and cycle embedding in some networks”, Frontiers of Mathematics in China, Vol. 4, pp.217-252, 2009.

M Arul Jeya Shalini, Jessie Abraham, Aswathi D, "Optimal Layout of Chord Graph into the Windmill Graph", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.55-58, 2019.

Odd graceful labeling is one of the major evolving research areas in the field of graph labeling. It is defined as for any graph G with q edges if there is an injection f from V(G) to {0, 1, 2, …, (2q-1)} such that, when each edge xy is assigned the label │f(x) ─ f(y)│,so that the edge labels are {1, 3, 5, …, (2q-1)} then the graph G is said to be odd graceful. Graph labeling has a vast range of real life applications which has provided major contributions in the development of new technologies. In this paper we have investigated and proved that the graph G which is obtained by joining m isomorphic copies of lobster graph to each vertex of the cycle Cm admits odd graceful labeling.

Graceful labeling, Odd graceful labeling, Cycle, Lobster

[1] C. Barrientos, Odd-graceful labelings, preprint.
[2] J. A.Gallian, Electronics Journal of Combinatorics, (2017).
[3] Gnanajothi R.B., Ph. D. Thesis, Madurai Kamaraj University,(1991).
[4] S. W. Golomb, How to number a graph, in Graph Theory and Computing, R. C. Read, ed., Academic Press, New York (1972).
[5] D Morgan – All lobsters with perfect matching are odd graceful, Electronic notes in Discrete Mathematics, (2002).
[6] M.I. Moussa, The International Journal on Application of Graph Theory in Wireless Ad hoc Networks,2(2010).
[7] M.I. Moussa, Some simple algorithm for odd graceful labeling graphs, proceed9th WSEAS Internat. Conf. Applied Informatics and Communications (AI `09) August, 2009, Moscow, Russia.
[8] M. I. Moussa and E. M. Badr, Odd graceful labelings of crown graphs, 1s Internat. Conf.
[9] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Pari (1967).
[10] Zhou, Yao, Chen and Tao a proof to the odd-gracefulness of all lobsters, Ars Combin., (2012).

J.JebaJesintha, Jayaglory.R, Bhabita G, "Odd Graceful Labelling of the Union of Cycle and Lobsters", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.59-63, 2019.

In this paper, we prove that the subdivision of Flower graph and the Path union of k copies of subdivision of Flower graphs are Product cordial graph and Total product cordial graphs. We also extend to prove that the path union from the outer vertex of the subdivision of Flower admits Product cordial and total product cordial labeling.

Product cordial labeling, Total product cordial labeling, Flower graph, Subdivision, Path union

[1] I.Cahit, Cordial graphs, A weaker version of graceful and harmonious graphs,ArsCombinatoria, Vol.23(1987), pp.201-207.
[2] Gallian J. A., A dynamical survey of graph labeling, Electronics Journal of Combinatorics,17th Ed.,(2017).
[3] Rosa.A, Theory of Graphs, Gordon and Breach, (1967).
[4] M. Sundaram, R. Ponraj and S.Somasundaram,(2004) Product cordial labeling of graphs,Bull. Pure and Appl. Sci. (Math. & Stat.), Vol. 23E pp.155-163
[5] M. Sundaram, R. Ponraj, and S. Somasundram, (2006) Total product cordial labeling of graphs, Bull.Pure Appl. Sci. Sect. E Math. Stat., Vol.25 pp.199-203.
[6] S. K. Vaidya and C. M. Barasara,(2011) Product cordial graphs in the context of some graph operations, Internal. J. Math. Sci. Comput., Vol.1(2).

Sharon Philomena. V, Priyadharshini. T, "On Variation of Product Cordial Labeling of Subdivision of Flower and Its Path Union", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.64-68, 2019.

The performance ability of a distributed multiprocessor is determined by its corresponding interconnection network and the primary criteria for choosing an appropriate interconnection network is its graph embedding capability. An embedding of a graph G into a graph H is an injective map on the vertices such that each edge of G is mapped into a shortest path of H. The wirelength of this embedding is the sum of the number of paths corresponding to G crossing every edge in H. In this paper we embed the locally twisted cube into rooted hypertrees to obtain the exact wirelength.

Emedding, locally twisted cube, rooted hypertree, wirelength

[1] Abraham, J., Arockiaraj, M., “Layout of embedding locally twisted cube into the extended theta mesh topology”, Vol 63, pp.371-379 , 2017.
[2] Abraham, J., Arockiaraj, M., “Wirelength of enhanced hypercubes into r-rooted complete binary trees”, Vol 53, pp.373-382 , 2016.
[3] Arockiaraj, M., Abraham, J., Quadras, J., Shalini, A.J.: “Linear layout of locally twisted cubes”. International Journal of Computer Mathematics. Vol 94, Issue.1, pp. 56-65, 2017
[4] Arockiaraj, M.,Rajasingh, I., Quadras, J., Shalini, A.J.: “Embedding hypercubes and folded hypercubes onto Cartesian product of certain trees”.Discrete Optimization. Vol 17, pp.1-13 ,2015
[5] Bezrukov, S.L., Chavez, J.D., Harper, L.H., Rottger, M., Schroeder, U.-P.: “Embedding of hypercubes into grids”, Mathematical Foundations of Computer Science, Springer Berlin Heidelberg, 1998.
[6] Bezrukov, S.L., Chavez, J.D., Harper, L.H., Rottger, M., Schroeder, U.-P.: “The congestion of n-cube layout on a rectangular grid”, Discrete Mathematics. Vol 213, Issue.1,pp.13-19, 2000
[7] Harper, L.H.: “Global Methods of Combinatorial Isoperimetric Problems”. Cambridge University Press, 2004.
[8] Lai Y.L., Williams, K.: “A survey of solved problems and applications on bandwidth, edge sum and profile of graphs”. Journal of Graph Theory. Vol 31,pp. 75-94, 1999
[9] Manuel, P., Arockiaraj, M., Rajasingh, I., Rajan, B.: “ Embedding hypercubes into
cylinders, snakes and caterpillars for minimizing wirelength”. Discrete Applied Mathematics. Vol 159, Issue .17,pp. 2109-2116, 2011
[10] Manuel, P., Rajasingh, I., Rajan, B., Mercy, H.: “Exact wirelength of hypercube on a
Grid”. Discrete Applied Mathematics. Vol 157, Issue 7, pp. 1486-1495, 2009
[11] Opatrny, J., Sotteau, D.: “Embeddings of complete binary trees into grids and extended
grids with total vertex-congestion “.Discrete Applied Mathematics.Vol 98,pp. 237-254 , 2000
[12] Rajan, R.S., Manuel, P., Rajasingh, I.: “Embeddings between hypercubes and
Hypertrees”. Journal of Graph Algorithms and Applications. Vol 19,Issue.1,pp. 361-373, 2015
[13] Yang, X., Evans, D.J., Megson, G.M.: “The Locally Twisted Cubes”. International
Journal of Computer Mathematics. Vol 82, Issue.4 pp.401-413, 2005

M Arul Jeya Shalini, Jessie Abraham, Sakthishwari S, "Exact Wirelength of Embedding Locally Twisted Cube into Rooted Hypertree", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.69-73, 2019.

A set D of vertices in a graph G=(V,E) is a nonsplit dominating set if the induced subgraph is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number of G and denoted γ_ns (G). In this paper, we define the nonsplit bondage number b_ns (G) of a graph G to be the minimum cardinality of a set E of edges for which γ_ns (G-E)> γ_ns (G). We obtain sharp bounds for b_ns (G) and obtain the exact values for some standard graphs.

Nonsplit dominating set, Nonsplit domination number, Bondage number, Nonsplit bondage number

[1] J. F. Fink, M. S. Jacobson, L. F. Kinch and J. Roberts, The Bondage number of a graph, Discrete Math. 86(1990) 47-57.
[2] F. Harary, Graph Theory, Addision-Wesley, Reading Mass. (1969).
[3] L. Hartnell, Douglas F. Rall, Bounds on the bondage number of a graph, Discrete Mathematics 128 (1994) 173-177.
[4] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Disc.Math., Marcel Dekker, Inc., New York, 1998.
[5] V. R. Kulli and B. Janakiram, The Nonsplit Domination Number of a Graph, Indian J.Pure appl. Math., 31(4):441-447, April 2000.

R. Jemimal Chrislight, Y. Therese Sunitha Mary, "The Nonsplit Bondage Number of Graphs", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.74-76, 2019.

A graph G(p, q) is said to be odd harmonious if there exists an injection f: V(G) → {0, 1, 2, …, 2q-1} such that the induced function f*:E(G)→{1,3,...,2q−1} defined by f * (uv) = f(u) + f(v) is a bijection. In this paper we prove that the subdivided shell graph, disjoint union of two subdivided shell graph, subdivided shell flower graph and subdivided uniform shell bow graph are odd harmonious.

disjoint union of graph, harmonious labeling, subdivided shell graph, odd harmonious labeling

[1] Deb P. and Limaye N.B., “On Harmonious Labeling of Some Cycle Related Graphs”, Ars Combina.,65(2002), 177-197.
[2] Jeba Jesintha J and Ezhilarasi Hilda K., “Subdivided Uniform Shell Bow Graphs are one Modulo three Graceful”, Mathematical Sciences International Research Journal, Vol.3, No.2(2014),645-646.
[3] Jeba Jesintha J and Ezhiarasi Hilda K.,Variation of “Graceful Labeling on Disjoint Union of two Subdivided Shell Graphs”, Annals of Pure and Applied Mathematics, Vol.8, No.2(2014), 19-25.
[4] Graham R. L. and Sloane N.J.A, “On Additive bases and Harmonious Graphs”, SIAM J. Algebr. Disc. Meth., 4(1980), 382–404.
[5] Harary F., “Graph Theory”, Addison-Wesley, Massachusetts, 1972.
[6] Liang Z., Bai Z., “On the Odd Harmonious Graphs with Applications”, J.Appl. Math.Comput., 29 (2009), 105–116.
[7] Jeyanthi P., Philo S. and Kiki A.Sugeng, “Odd Harmonious Labeling of Some New Families of Graph”s, SUT Journal of Mathematics,Vol.51, No.2(2015), 53-65.
[8] Jeyanthi P., Philo S., “Odd Harmonious Labeling of Some Cycle Related Graphs”, Proyecciones Journal of Mathematics, Vol.35, No.1, (2016)85-98.
[9] Jeyanthi P., Philo S, “Odd harmonious Labeling of Plus Graphs”, Bulletin of the International Mathematical Virtual Institute,Vol.7(2017),515-526.
[10] Jeyanthi P., Philo S., Siddiqui M.K., “Odd Harmonious Labeling of Super Subdivision Graphs”, Proyecciones Journal of Mathematics, Vol.38 No.1(2019), to appear.
[11]Jeyanthi P., Philo S., Maged Z Youssef., “Odd harmonious Labeling of Grid Graphs”, Proyecciones Journal of Mathematics, to appear.
[12]Jeyanthi P., Philo S., “Odd Harmonious Labeling of Some New Graphs”, Southeast Asian Bulletin of Mathematics, to appear.
[13]Jeyanthi P., Philo S., “Odd Harmonious Labeling of Step Ladder Graphs”, Utilitas Mathematica, to appear.

P.Jeyanthi, S. Philo, "Odd Harmonious Labeling of Subdivided Shell Graphs", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.77-80, 2019.

A Human chain graph is a simple, finite and undirected geaph. In this paper, we prove that the existence of V-cordial and Homo-cordial labeling for the Human chain graph by using algorithms.

Human chain, Cordial, V-cordial, Homo-cordial, Labeling

[1] K. Anitha, B. Selvam, “Human chain graph”, International journal of Engineering, Science and Mathematics, Vol. 7, Issue 8, 2018.
[2] Cahit, “On cordial and 3-equitable labelings of graphs”, Util. Math., Vol. 37, pp. 189198, 1990.
[3] J.A. Gallian, “A Dynamic Survey of graph labeling”, the Electronic Journal of combinatories, 19, # DS6, 2012.
[4] A. Nellai Murugan, P. Iyadurai Selvaraj, “Path Related Cup Cordial Graphs”, Indian Journal of Applied Research. Vol. 4, Issue 8, pp. 433436, 2011.
[5] A. Nellai Murugan, A. Madhubala, “Path Related Homo-Cordial Graphs”, International Journal of Innovative Science Engineering and Technology, Vol. 2, Issue 8, 2015.
[6] A. Nellai Murugan, A. Madhubala, “Special case of Homo-Cordial Graphs”, International Journal of Emerging Technologies in Engineering research, Vol. 2, Issue 3, pp. 15, 2015.
[7] Selvam, Avadayappan, M. Bhuvaneshwari, M. vasanthi, “Homo Cordial Graphs”, International Journal of Scientific Research, Vol. 5, Issue 5, pp. 700724, 2016.

K. Anitha, B. Selvam, K. Thirusangu, "Some Cordial Labeling on Human Chain Graph", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.81-84, 2019.

In this paper, we prove the existence of proper d – lucky labeling of the arbitrary super subdivision of some new family of graphs (〖 P〗_(m ) ∶ Q_3 )and [〖 P〗_(m ) : C_n^((2))]graphs and their proper d- lucky numbers are obtained.

Proper d-lucky labeling, proper d-lucky number, arbitrary super subdivision

[1] D. Ahima Emilet and Indra Rajasingh, “d- lucky labeling of cycle of ladder, n- sunlet and Helm Graphs”, International Journal of Pure and Applied Mathematics, Vol. 109, No. 10, pp. 219-227, 2016.
[2] E. Esakkiammal, K. Thirusangu, S. Seethalakshmi, “d-lucky labeling of Arbitrary Super Subdivision of Some Graphs”, International Journal of Pure and Applied Mathematics, Vol. 113, No. 7, pp. 93-101, 2017.
[3] E. Esakkiammal, K. Thirusangu, S. Seethalakshmi, “Proper d-lucky labeling of Arbitrary Super Subdivision of Some New family of Star and Cycle Graphs”, (communicated…)
[4] F. Harary, “Graph Theory”, Addison Wesley Reading, Massachusetts, 1972.
[5] Joseph A. Gallian, “A Dynamic Survey of Graph Labeling”, The Electronic Journal of Combinatorics, #DS6, 2017.
[6] Kins. Yenoke, R.C. Thivyarathi, D. Antony Xavier, “Proper Lucky Number of Mesh and its derived architectures”, Journal of Computer and Mathematical sciences, Vol. 8, Issue 5, pp. 187-195, 2017.
[7] Mirka Miller, Indra Rajasingh, D. Ahima Emilet, D. Azubha Jemilet, “d-lucky labeling of graphs”, Procedia Computer Science Vol. 57, pp. 766-771, 2015.
[8] Selvam Avadayappan, R. Vasuki, “New Families of Mean Graphs”, International J. Math Combin, Vol. 2, pp. 68-80, 2010.

E. Esakkiammal, K. Thirusangu, S. Seethalakshmi, "Proper D - Lucky Labeling on Arbitrary Super Subdivision of New Family of Graphs", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.85-90, 2019.

Finding a partition V_1,V_2,...,V_k of V(G) with minimum k is called the induced H-packing k-partition problem of G. The minimum induced H-packing k-partition number is denoted by ipp(G,H). In this paper we determine an induced P_3-packing k-partition number for Butterfly Networks, Honeycomb Networks, and Circum Pyrene with H is isomorphic to P_3.

Perfect P_3-packing, Almost Perfect P_3-packing, Induced H-packing k-partition, Butterfly networks, Honeycomb Networks and Circum Pyrene

[1] A. Al Mutairi, Bader Ali and D. Paul Manuel, "Packing in Carbon Nanotubes", Journal of Combinatorial Mathematics and Combinatorial Computing , 92, 2015, 195 - 206.
[2] R. Bar-Yehuda, M. Halldorsson, J. Naor, H. Shachnai and I. Shapira, "Scheduling split intervals", in: Proc. Thirteenth Annu. ACM - SIAM Symp, 2002,732 - 741.
[3] R. Bejar, B. Krishnamachari, C. Gomes and B. Selman, "Distributed constraint satisfaction in a wireless sensor tracking system", Workshop on Distributed Constraint Reasoning, Internat. Joint Conf. on Artificial Intelligence, 2001.
[4]A. Felzenbaum, "Packing lines in a hypercube", Discrete Mathematics 117, 1993, 107 - 112.
[5] P. Hell and D. Kirkpatrick, "On the complexity of a generalized matching problem", in: Proc. Tenth ACM Symp, 1978, 309 - 318.
[6] A. Muthumalai, I. Rajasingh and A. S. Shanthi, "Packing of Hexagonal Networks", Journal of Combinatorial Mathematics and Combinatorial Computation, 79, 2011, 121 - 127.
[7] D. Paul Manuel, I. Mostafa Abd-El-Barr, I. Rajasingh, Bharati Rajan, "An efficient representation of Benes networks and its applications", Journal of Discrete Algorithms, 6, 2008, 11 - 19.
[8] J. Quadras, K. Balasubramanian, K.A. Christy, "Analytical expressions for Wiener indices of n-circumscribed peri-condensed benzenoid graphs", Journal of Mathematical Chemistry, 54, 2016, 823 - 843.
[9] S.M.J. Raja, A. Xavier, I. Rajasingh, "Induced H-packing k-partition problem in interconnection networks", International Journal of Computer Mathematics: Computer Systems Theory. 2, 2017 136 - 146.
[10] I. Rajasingh, A. Muthumalai, R. Bharati and A. S. Shanthi, "Packing in honeycomb networks", Journal of Mathematical Chemistry, 50, 5, 2012, 1200 - 1209.
[11] H. Liu, L. Xie, J. Liu, L. Ding, "Application of Butterfly Clos-Network in Network-on-Chip", The Scientific World Journal, 2014, 1–11.

Antony Xavier, Santiagu Theresal, S. Maria Jesu Raja, "Induced P_3-Packing k-partition Number for Certain Graphs", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.91-95, 2019.

The concept of convex sets in the classical Euclidean geometry was extended to graphs and different graph convexities were studied based on the kind of path that is considered. The geodetic number of a graph is one of the extensively studied graph theoretic parameters concerning geodesic convexity in graphs. A u-v geodesic is a u-v path of length d(u,v) in G. For a non-trivial connected graph G , a set S ⊆ V (G) is called a geodetic set if every vertex not in S lies on a geodesic between two vertices from S. The cardinality of the minimum geodetic set of G is the geodetic number g(G) of G. The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. In this paper some of the geodetic variants including hull number, monophonic hull number, geodetic number, strong geodetic number , total geodetic number ,upper geodetic number, open geodetic number and strong open geodetic number for Sierpinski triangle is investigated.

geodetic number, strong geodetic number, total geodetic number, hull number

[1] Ahangar, H. Abdollahzadeh, and Vladimir Samodivkin. "The total geodetic number of a graph." Util. Math 100 (2016): 253-268.
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[3] Chartrand, Gary, Frank Harary, and Ping Zhang. "On the geodetic number of a graph." Networks39.1 (2002): 1-6.
[4] Chartrand, Gary, et al. "Geodominationin graphs." Bulletin of the Institute of Combinatorics and its Applications 31 (2001): 51-59.
[5] Dourado, Mitre C., et al. "Some remarks on the geodetic number of a graph." Discrete Mathematics 310.4 (2010): 832-837.
[6] Everett, Martin G., and Stephen B. Seidman. "The hull number of a graph." Discrete Mathematics 57.3 (1985): 217-223.
[7] Harary, Frank, Emmanuel Loukakis, and Constantine Tsouros. "The geodetic number of a graph." Mathematical and Computer
[8] Hansberg, Adriana, Lutz Volkmann, “On the geodetic and geodetic domination numbers of a graph,” Discrete Mathematics, vol.310, no. 15, pp.2140-2146, 2010.
[9] Iršič, Vesna. "Strong geodetic number of complete bipartite graphs and of graphs with specified diameter." Graphs and Combinatorics 34.3 (2018): 443-456.
[10] Jakovac, Marko, and Sandi Klavžar. "Vertex-, edge-, and total-colorings of Sierpiński-like graphs." Discrete Mathematics309.6 (2009): 1548-1556.
[11] Jakovac, Marko. "A 2-parametric generalization of Sierpinski gasket graphs." Ars Comb. 116 (2014): 395-405.
[12] John, J., and V. Mary Gleeta. "The Forcing Monophonic Hull Number of a Graph." International Journal of Mathematics Trends and Technology 3.2 (2012): 43-46.
[13] Manuel, Paul, et al. "Strong geodetic problem in networks: computational complexity and solution for Apollonian networks." arXiv preprint arXiv:1708.03868 (2017).
[14] Pelayo, Ignacio M., Geodesic convexity in graphs, Springer, New York, 2013.Modelling 17.11 (1993): 89-95.
[15] Parisse, Daniele. "Sierpinski Graphs S (n, k)." Ars Combinatoria90 (2009): 145-160.
[16] Rodríguez-Velázquez, Juan Alberto, Erick David Rodríguez-Bazan, and Alejandro Estrada-Moreno. "On generalized Sierpiński graphs." Discussiones Mathematicae Graph Theory37.3 (2017): 547-560.
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[18] A. P. Santhakumaran and T. Kumari Latha, On the open geodetic number of a graph, SCIENTIA, Series A :Mathematical Sciences, 19 (2010),131-142.
[19] Xavier, D. Antony, S. ARUL AMIRTHA RAJA"Strong Open geodetic Set in Graphs”(Communicated)

Deepa Mathew, D. Antony Xavier, "Geodetic Variants of Sierpinski Triangles", International Journal of Computer Sciences and Engineering, Vol.07, Issue.05, pp.96-100, 2019.