Graphs with Bounded Induced Distance

  • Serafino Cicerone
  • Gabriele Di Stefano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)


In this work we introduce graphs with bounded induced distance of order k (BID(k) for short). In any graph belonging to BID(k), the length of every induced path between every pair of nodes is at most k times the distance between the same nodes. In communication networks modeled by these graphs any message can be always delivered through a path whose length is at most k times the best possible one, even if some nodes fail.

In this work we first provide a characterization of graphs in BID(k) by means of cycle-chord conditions. After that, we investigate classes with order k ≤ 2. In this context, we note that the class BID(1) is the well known class of distance-hereditary graphs, and show that 3/2 is a lower bound for the order k of graphs that are not distance-hereditary. Then we characterize graphs in BID(2/3) by means of their minimal forbidden induced subgraphs, and we also show that graphs in BID(2) have a more complex characterization. We prove that the recognition problem for the generic class BID(k) is Co-NP-complete. Finally, we show that the split composition can be used to generate graphs in BID(k).


Short Path Steiner Tree Recognition Problem Node Failure Graph Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bandelt, H.J., D’Atri, A., Moscarini, M., Mulder, H.M., Schultze, A.: Operations on distance hereditary graphs. Technical Report 226, CNR, Istituto di Analisi dei Sistemi e Informatica del CNR, Rome Italy (1988)Google Scholar
  2. 2.
    Bandelt, H.J., Mulder, M.: Distance-hereditary graphs. Journal of Combinatorial Theory, Series B 41(2), 182–208 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouchet, A.: Transforming trees by successive local complementations. Journal of Graph Theory 4, 196–207 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A.: Special graph classes. Technical Report SM-DU-199, University Duisburg (1993)Google Scholar
  5. 5.
    Brandstädt, A., Dragan, F.F.: A linear time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs. Technical Report SM-DU-261, University Duisburg (1994)Google Scholar
  6. 6.
    Cicerone, S., Di Stefano, G.: Graph classes between parity and distance-hereditary graphs. In: 1st Discrete Mathematics and Theoretical Computer Science (DMTCS 1996), Combinatorics, Complexity, and Logic, pp. 168–181. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Cicerone, S., Di Stefano, G.: Port and node support to compact routing. Technical Report R.97-17, Dipartimento di Ingegneria Elettrica, Universitá di L’Aquila, L’Aquila, Italy (1997)Google Scholar
  8. 8.
    Cunningham, W.H.: Decomposition of directed graphs. SIAM Journal on Alg. Disc. Meth. 3, 214–228 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Atri, A., Moscarini, M.: Distance-hereditary graphs, steiner trees, and connected domination. SIAM Journal on Computing 17, 521–530 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Di Stefano, G.: A routing algorithm for networks based on distance-hereditary topologies. In: 3rd Int. Colloquium on Structural Information and Communication Complexity, SIROCCO 1996 (1996)Google Scholar
  11. 11.
    Dragan, F.F.: Dominating cliques in distance-hereditary graphs. In: Schmidt, E.M., Skyum, S. (eds.) SWAT 1994. LNCS, vol. 824, pp. 370–381. Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Esfahanian, A.H., Oellermann, O.R.: Distance-hereditary graphs and mult-idestination message-routing in multicomputers. Journal of Comb. Math. and Comb. Computing 13, 213–222 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Frederickson, G.N., Janardan, R.: Designing networks with compact routing tables. Algorithmica 3, 171–190 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Hammer, P.L., Maffray, F.: Completely separable graphs. Discrete Applied Mathematics 27, 85–99 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)zbMATHGoogle Scholar
  17. 17.
    Howorka, E.: Distance hereditary graphs. Quart. J. Math. Oxford 2(28), 417–420 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    McQuillan, J.M., Richer, I., Rosen, E.C.: The new routing algorithm for the ARPANET. IEEE Trans. Commun. COM 28, 711–719 (1980)CrossRefGoogle Scholar
  19. 19.
    Merlin, P.M., Segall, A.: A failsafe distributed routing protocol. IEEE Trans. Commun. COM 27, 1280–1289 (1979)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nicolai, F.: Hamiltonian problems on distance-hereditary graphs. Technical Report SM-DU-264, University Duisburg (1994)Google Scholar
  21. 21.
    Peleg, D., Schaffer, A.: Graph spanners. Journal of Graph Theory 13, 99–116 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schwartz, M., Stern, T.E.: Routing techniques used in computer communication networks. IEEE Trans. Commun. COM 28, 539–552 (1980)CrossRefGoogle Scholar
  23. 23.
    van Leeuwen, J., Tan, R.B.: Interval routing. The Computer Journal 30, 298–307 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yeh, H.G., Chang, G.J.: Weighted connected domination and steiner trees in distance-hereditary graphs. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 48–52. Springer, Heidelberg (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Serafino Cicerone
    • 1
  • Gabriele Di Stefano
    • 1
  1. 1.Dipartimento di Ingegneria ElettricaUniversità degli Studi di L’AquilaMonteluco di Roio, L’AquilaItaly

Personalised recommendations