Boundary Vertices of Cartesian Product of Directed Graphs

  • Manoj Changat
  • Prasanth G. Narasimha-Shenoi
  • Mary Shalet Thottungal JosephEmail author
  • Ram Kumar
Original Paper


Suppose that \(D=(V,E)\) is a strongly connected digraph. Let \(u, v \in V(D)\). The maximum distance md(uv) defined as \(md(u,v)=max\{\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)\}\) is a metric, where \(\overrightarrow{d}(u,v)\) denote the length of a shortest directed \(u-v\) path in D. The boundary, contour, eccentricity and periphery sets of a strong digraph D are defined with respect to this metric. The main aim of this paper is to identify the above said metrically defined sets of a strong digraph D in terms of its prime factor decomposition with respect to Cartesian product.


Maximum distance Boundary Contour Eccentricity Periphery Two sided eccentricity property 

Mathematics Subject Classification




Prasanth G. Narasimha-Shenoi and Mary Shalet T. J. are supported by Science and Engineering Research Board, a statutory board of Government of India under their Extra Mural Research Funding No. EMR/2015/002183. Also, their research was partially supported by Kerala State Council for Science Technology and Environment of Government of Kerala under their SARD project grant Council(P) No. 436/2014/KSCSTE. Authors acknowledge the reviewers for their valuable comments and suggestions.


  1. 1.
    Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, Algorithms and Applications. Springer, Berlin (2008)zbMATHGoogle Scholar
  2. 2.
    Brešar, B., Klavžar, S., Horvat, A.T.: On the geodetic number and related metric sets in cartesian product graphs. Discrete Math. 308(23), 5555–5561 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cáceres, J., Márquez, A., Oellermann, O.R., Puertas, M.L.: Rebuilding convex sets in graphs. Discrete Math. 297(1), 26–37 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C.: On geodetic sets formed by boundary vertices. Discrete Math. 306(2), 188–198 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chartrand, G., Tian, S.: Distance in digraphs. Comput. Math. Appl. 34(11), 15–23 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chartrand, G., Erwin, D., Johns, G.L., Zhang, P.: Boundary vertices in graphs. Discrete Math. 263(1), 25–34 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Crespelle, C., Thierry, E., Lambert, T.: A linear-time algorithm for computing the prime decomposition of a directed graph with regard to the Cartesian product. In: International computing and combinatorics conference, pp. 469–480. Springer (2013)Google Scholar
  8. 8.
    Feigenbaum, J.: Directed Cartesian-product graphs have unique factorizations that can be computed in polynomial time. Discrete Appl. Math. 15(1), 105–110 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Harary, F., Trauth Jr., C.A.: Connectedness of products of two directed graphs. SIAM J. Appl. Math. 14(2), 250–254 (1966)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Imrich, W., Klavzar, S.: Product Graphs: Structure and Recognition. Wiley, Hoboken (2000)zbMATHGoogle Scholar
  11. 11.
    Nebeskỳ, L.: The directed geodetic structure of a strong digraph. Czechoslov. Math. J. 54(1), 1–8 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
  13. 13.
    Robbins, H.: A theorem on graphs, with an application to a problem of traffic control. Am. Math. Mon. 46(5), 281–283 (1939)CrossRefGoogle Scholar
  14. 14.
    Sabidussi, G.: Graph multiplication. Math. Z. 72(1), 446–457 (1959)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  • Manoj Changat
    • 1
  • Prasanth G. Narasimha-Shenoi
    • 2
  • Mary Shalet Thottungal Joseph
    • 2
    Email author
  • Ram Kumar
    • 3
  1. 1.Department of Futures StudiesUniversity of KeralaTrivandrumIndia
  2. 2.Department of MathematicsGovernment College ChitturPalakkadIndia
  3. 3.Department of MathematicsM. G. CollegeTrivandrumIndia

Personalised recommendations