Skip to main content
SpringerLink
Log in
Book cover

Geodesic Convexity in Graphs pp 91–93Cite as

Computational Complexity

  • Chapter
  • First Online:

  • 1041 Accesses

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

Abstract

Let G be a graph of order n and size m and S a proper subset of V (G). The geodetic closure I[S] can be obtained in O( | S | m) time by applying | S | times Bread First Search, each one starting from a distinct s ∈ S. In particular, the problem of verifying if S is convex can be done in O( | S | m) time. By iteratively applying this process, the convex hull [S] can be determined in O([S]m) time [81].

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Anand, B.S., Changat, M., Klavžar, S., Peterin, I.: Convex sets in lexicographic products of graphs. Graphs Combin. 28, 77–84 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Aniversario, I.S., Jamil, F.P., Canoy, S.R.: The closed geodetic numbers of graphs. Util. Math. 74, 3–18 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Araujo, J., Campos, V., Giroire, F., Nisse, N., Sampaio, L., Soares, R.: On the hull number of some graph classes. Theor. Comput. Sci. 475, 1–12 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Artigas, D., Dantas, S., Dourado, M.C., Szwarcfiter, J.L.: Partitioning a graph into convex sets. Discrete Math. 311(17), 1968–1977 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Artigas, D., Dantas, S., Dourado, M.C., Szwarcfiter, J.L., Yamaguchi, S.: On the contour of graphs. Discrete Appl. Math. 161(10–11), 1356–1362 (2013)

    MathSciNet  Google Scholar 

  6. Atici, M.: Computational complexity of geodetic set. Int. J. Comput. Math. 79(5), 587–591 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Atici, M.: On the edge geodetic number of a graph. Int. J. Comput. Math. 80(7), 853–861 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Atici, M., Vince, A.: Geodesics in graphs, an extremal set problem and perfect Hash families. Graphs Combin. 18(3), 403–413 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Bandelt, H.-J., Chepoi, V.: A Helly theorem in weakly modular spaces. Discrete Math. 125, 25–39 (1996)

    MathSciNet  Google Scholar 

  10. Bandelt, H.-J., Chepoi, V.: Metric graph theory and geometry: a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry: Twenty Years Later. Contemporary Mathematics, vol. 453, pp. 49–86. American Mathematical Society, Providence (2008)

    Google Scholar 

  11. Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41(2), 182–208 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Barbosa, R.M., Coelho, E.M.M., Dourado, M.C., Rautenbach, D., Szwarcfiter, J.L.: On the Carathéodory number for the convexity of paths of order three. SIAM J. Discrete Math. 26(3), 929–939 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Bermudo, S., Rodríguez-Velázquez, J.A., Sigarreta, J.M., Yero, I.G.: On geodetic and k-geodetic sets in graphs. Ars Combin. 96, 469–478 (2010)

    Google Scholar 

  14. Brandstädt, A., V.B., Spinrad, J.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)

    Google Scholar 

  15. Brešar, B., Gologranc, T.: On a local 3-Steiner convexity. European J. Combin. 32(8), 1222–1235 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Brešar, B., Tepeh Horvat, A.: On the geodetic number of median graphs. Discrete Math. 308(18), 4044–4051 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Brešar, B., Klavžar, S., Tepeh Horvat, A.: On the geodetic number and related metric sets in Cartesian product graphs. Discrete Math. 308(23), 5555–5561 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Brešar, B., Changat, M., Mathews, J., Peterin, I., Narasimha-Shenoi, P.G., Tepeh Horvat, A.: Steiner intervals, geodesic intervals, and betweenness. Discrete Math. 309(20), 6114–6125 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Brešar, B., Kovše, M., Tepeh Horvat, A.: Geodetic sets in graphs. In: Structural Analysis of Complex Networks, pp. 197–218. Birkhäuser/Springer, New York (2011)

    Google Scholar 

  20. Brešar, B., Šumenjak, T.K., Tepeh Horvat, A.: The geodetic number of the lexicographic product of graphs. Discrete Math. 311(16), 1693–1698 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Redwood City (1990)

    MATH  Google Scholar 

  22. Buckley, F., Harary, F., Quintas, L.V.: Extremal results on the geodetic number of a graph. Scientia 2A, 17–26 (1988)

    MathSciNet  Google Scholar 

  23. Cáceres, J., Márquez, A., Oellermann, O.R., Puertas, M.L.: Rebuilding convex sets in graphs. Discrete Math. 297(1–3), 26–37 (2005)

    MathSciNet  MATH  Google Scholar 

  24. 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)

    MathSciNet  MATH  Google Scholar 

  25. Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C.: Geodeticity of the contour of chordal graphs. Discrete Appl. Math. 156(7), 1132–1142 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Cáceres, J., Márquez, A., Puertas, M.L.: Steiner distance and convexity in graphs. European J. Combin. 29(3), 726–736 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L.: On the geodetic and the hull numbers in strong product graphs. Comput. Math. Appl. 60(11), 3020–3031 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Cáceres, J., Oellermann, O.R., Puertas, M.L.: Minimal trees and monophonic convexity. Discuss. Math. Graph Theory 32(4), 685–704 (2012)

    MathSciNet  Google Scholar 

  29. Cagaanan, G.B., Canoy, S.R.: On the hull sets and hull number of the Cartesian product of graphs. Discrete Math. 287, 141–144 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Cagaanan, G.B., Canoy, S.R.: On the geodetic covers and geodetic bases of the composition G[K m ]. Ars Combin. 79, 33–45 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Cagaanan, G.B., Canoy, S.R.: Bounds for the geodetic number of the Cartesian product of graphs. Util. Math. 79, 91–98 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Calder, J.R.: Some elementary properties of interval convexities. J. London Math. Soc. 3(2), 422–428 (1971)

    MathSciNet  MATH  Google Scholar 

  33. Canoy, S.R., Cagaanan, G.B.: On the geodesic and hull numbers of the sum of graphs. Congr. Numer. 161, 97–104 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Canoy, S.R., Cagaanan, G.B.: On the hull number of the composition of graphs. Ars Combin. 75, 113–119 (2005)

    MathSciNet  MATH  Google Scholar 

  35. Canoy, S.R., Garces, I.J.L.: Convex sets under some graph operations. Graphs Combin. 18(4), 787–793 (2002)

    MathSciNet  MATH  Google Scholar 

  36. Canoy, S.R., Cagaanan, G.B., Gervacio, S.V.: Convexity, geodetic and hull numbers of the join of graphs. Util. Math. 71, 143–159 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Centeno, C.C., Dantas, S., Dourado, M.C., Rautenbach, D., Szwarcfiter, J.L.: Convex partitions of graphs induced by paths of order three. Discrete Math. Theory Comput. Sci. 12(5), 175–184 (2010)

    MathSciNet  Google Scholar 

  38. Chae, G.-B., Palmer, E.M., Siu, W.-C.: Geodetic number of random graphs of diameter 2. Australas. J. Combin. 26, 11–20 (2002)

    MathSciNet  MATH  Google Scholar 

  39. Chang, G.J., Tong, L.-D., Wang, H.-T.: Geodetic spectra of graphs. European J. Combin. 25(3), 383–391 (2004)

    MathSciNet  MATH  Google Scholar 

  40. Changat, M., Mathew, J.: On triangle path convexity in graphs. Discrete Math. 206, 91–95 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Changat, M., Mathew, J.: Induced path transit function, monotone and Peano axioms. Discrete Math. 286(3), 185–194 (2004)

    MathSciNet  MATH  Google Scholar 

  42. Changat, M., Klavžar, S., Mulder, H.M.: The all-paths transit function of a graph. Czechoslovak Math. J. 51(2), 439–448 (2001)

    MathSciNet  MATH  Google Scholar 

  43. Changat, M., Mulder, H.M., Sierksma, G.: Convexities related to path properties on graphs. Discrete Math. 290(2–3), 117–131 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Changat, M., Mathew, J., Mulder, H.M.: The induced path function, monotonicity and betweenness. Discrete Appl. Math. 158(5), 426–433 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Changat, M., Narasimha-Shenoi, P.G., Mathews, J.: Triangle path transit functions, betweenness and pseudo-modular graphs. Discrete Math. 309(6), 1575–1583 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Changat, M., Narasimha-Shenoi, P.G., Pelayo, I.M.: The longest path transit function of a graph and betweenness. Util. Math. 82, 111–127 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Changat, M., Lakshmikuttyamma, A.K., Mathews, J., Peterin, I., Prasanth, N.-S., Tepeh, A.: A note on 3-Steiner intervals and betweenness. Discrete Math. 311(22), 2601–2609 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Chartrand, G., Zhang, P.: Convex sets in graphs. Congr. Numer. 136, 19–32 (1999)

    MathSciNet  MATH  Google Scholar 

  49. Chartrand, G., Zhang, P.: The forcing geodetic number of a graph. Discuss. Math. Graph Theory 19, 45–58 (1999)

    MathSciNet  MATH  Google Scholar 

  50. Chartrand, G., Zhang, P.: The geodetic number of an oriented graph. Europ. J. Combin. 21, 181–189 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Chartrand, G., Zhang, P.: On graphs with a unique minimum hull set. Discuss. Math. Graph Theory 21(1), 31–42 (2001)

    MathSciNet  MATH  Google Scholar 

  52. Chartrand, G., Zhang, P.: The forcing convexity number of a graph. Czechoslovak Math. J. 51(4), 847–858 (2001)

    MathSciNet  MATH  Google Scholar 

  53. Chartrand, G., Zhang, P.: The forcing hull number of a graph. J. Combin. Math. Combin. Comput. 36, 81–94 (2001)

    MathSciNet  MATH  Google Scholar 

  54. Chartrand, G., Zhang, P.: Extreme Geodesic Graphs. Czechoslovak Math. J. 52(4), 771–780 (2002)

    MathSciNet  MATH  Google Scholar 

  55. Chartrand, G., Zhang, P.: The Steiner number of a graph. Discrete Math. 242, 41–54 (2002)

    MathSciNet  MATH  Google Scholar 

  56. Chartrand, G., Oellermann, O.R., Tian, S., Zou, H.: Steiner distance in graphs. Časopis Pěst. Mat. 114(4), 399–410 (1989)

    MathSciNet  MATH  Google Scholar 

  57. Chartrand, G., Zhang, P., Harary, F.: Extremal problems in geodetic graph theory. Cong. Numer. 131, 55–66 (1998)

    MathSciNet  MATH  Google Scholar 

  58. Chartrand, G., Harary, F., Zhang, P.: Geodetic sets in graphs. Discuss. Math. Graph Theory 20, 129–138 (2000)

    MathSciNet  MATH  Google Scholar 

  59. Chartrand, G., Harary, F., Zhang, P.: On the hull number of a graph. Ars Combin. 57, 129–138 (2000)

    MathSciNet  MATH  Google Scholar 

  60. Chartrand, G., Harary, F., Swart, H.C., Zhang, P.: Geodomination in graphs. Bull. Inst. Combin. Appl. 31, 51–59 (2001)

    MathSciNet  MATH  Google Scholar 

  61. Chartrand, G., Chichisan, A., Wall, C.E., Zhang, P.: On convexity in graphs. Congr. Numer. 148, 33–41 (2001)

    MathSciNet  MATH  Google Scholar 

  62. Chartrand, G., Fink, J.F., Zhang, P.: Convexity in oriented graphs. Discrete Math. 116(1–2), 115–126 (2002)

    MathSciNet  MATH  Google Scholar 

  63. Chartrand, G., Harary, F., Zhang, P.: On the geodetic number of a graph. Networks 39, 1–6 (2002)

    MathSciNet  MATH  Google Scholar 

  64. Chartrand, G., Wall, C.E., Zhang, P.: The convexity number of a graph. Graphs and Combin. 18(2), 209–217 (2002)

    MathSciNet  MATH  Google Scholar 

  65. Chartrand, G., Palmer, E.M., Zhang, P.: The geodetic number of a graph: a survey. Congr. Numer. 156, 37–58 (2002)

    MathSciNet  MATH  Google Scholar 

  66. Chartrand, G., Erwin, D., Johns, G.L., Zhang, P.: Boundary vertices in graphs. Discrete Math. 263, 25–34 (2003)

    MathSciNet  MATH  Google Scholar 

  67. Chartrand, G., Fink, J.F., Zhang, P.: The hull number of an oriented graph. Int. J. Math. Math. Sci. 36, 2265–2275 (2003)

    MathSciNet  Google Scholar 

  68. Chartrand, G., Garry, L., Zhang, P.: The detour number of a graph. Util. Math. 64, 97–113 (2003)

    MathSciNet  MATH  Google Scholar 

  69. Chartrand, G., Garry, L., Zhang, P.: On the detour number and geodetic number of a graph. Ars Combin. 72, 3–15 (2004)

    MathSciNet  MATH  Google Scholar 

  70. Chartrand, G., Escuadro, H., Zhang, P.: Detour distance in graphs. J. Combin. Math. Combin. Comput. 52, 75–94 (2005)

    MathSciNet  Google Scholar 

  71. Chartrand, G., Lesniak, L., Zhang, P.: Graphs and Digraphs, 5th edn. CRC Press, Boca Raton (2011)

    Google Scholar 

  72. Chastand, M., Polat, N.: On geodesic structures of weakly median graphs I. Decomposition and octahedral graphs. Discrete Math. 306(13), 1272–1284 (2006)

    MathSciNet  MATH  Google Scholar 

  73. Chastand, M., Polat, N.: On geodesic structures of weakly median graphs II: Compactness, the role of isometric rays. Discrete Math. 306(16), 1846–1861 (2006)

    MathSciNet  MATH  Google Scholar 

  74. Chepoi, V.: Isometric subgraphs of Hamming graphs and d-convexity. Cyber. Syst. Anal. 24(1), 6–11 (1988)

    MathSciNet  Google Scholar 

  75. Chepoi, V.: Separation of two convex sets in convexity structures. J. Geom. 50(1–2), 30–51 (1994)

    MathSciNet  MATH  Google Scholar 

  76. Chepoi, V.: Peakless functions on graphs. Discrete Appl. Math. 73(2), 175–189 (1997)

    MathSciNet  MATH  Google Scholar 

  77. Cyman, J., Lemanska, M., Raczek, J.: Graphs with convex domination number close to their order. Discuss. Math. Graph Theory 26(2), 307–316 (2006)

    MathSciNet  MATH  Google Scholar 

  78. Dirac, G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)

    MathSciNet  MATH  Google Scholar 

  79. Dong, L., Lu, C., Wang, X.: The upper and lower geodetic numbers of graphs. Ars Combin. 91, 401–409 (2009)

    MathSciNet  MATH  Google Scholar 

  80. Dourado, M.C., Protti, F., Szwarcfiter, J.L.: On the complexity of the geodetic and convexity numbers of a graph. RMS Lect. Notes Ser. Math. 7, 101–108 (2008)

    MathSciNet  Google Scholar 

  81. Dourado, M.C., Gimbel, J.G., Kratochvíl, J., Protti, F., Szwarcfiter, J.L.: On the computation of the hull number of a graph. Discrete Math. 309(18), 5668–5674 (2009)

    MathSciNet  MATH  Google Scholar 

  82. Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: On the hull number of triangle-free graphs. SIAM J. Discrete Math. 23(4), 2163–2172 (2009/10)

    Google Scholar 

  83. Dourado, M.C., Protti, F., Szwarcfiter, J.L.: Complexity results related to monophonic convexity. Discrete Appl. Math. 158(12), 1268–1274 (2010)

    MathSciNet  MATH  Google Scholar 

  84. Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: Some remarks on the geodetic number of a graph. Discrete Math. 310, 832–837 (2010)

    MathSciNet  MATH  Google Scholar 

  85. Dourado, M.C., Rautenbach, D., dos Santos, V.F., Schäfer, P.M., Szwarcfiter, J.L., Toman, A.: An upper bound on the P 3-Radon number. Discrete Math. 312(16), 2433–2437 (2012)

    MathSciNet  MATH  Google Scholar 

  86. Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: On the convexity number of graphs. Graphs Combin. 28(3), 333–345 (2012)

    MathSciNet  MATH  Google Scholar 

  87. Dourado, M.C., Rautenbach, D., de Sá, V.G.P., Szwarcfiter, J.L.: On the geodetic radon number of grids. Discrete Math. 313(1), 111–121 (2013)

    MathSciNet  MATH  Google Scholar 

  88. Dragan, F.F., Nicolai, F., Brandstädt, A.: Convexity and HHD-free graphs. SIAM J. Discrete Math. 12, 119–135 (1999)

    MathSciNet  MATH  Google Scholar 

  89. Duchet, P.: Convex sets in graphs II. Minimal path convexity. J. Comb. Theory Ser. B 44(3), 307–316 (1988)

    MathSciNet  MATH  Google Scholar 

  90. Duchet, P.: Convexity in combinatorial structures. Rend. Circ. Mat. Palermo 14(2), 261–293 (1987)

    Google Scholar 

  91. Duchet, P.: Discrete convexity: retractions, morphisms and the partition problem. Proceedings of the Conference on Graph Connections, India, 1998, pp. 10–18. Allied Publishers, New Delhi (1999)

    Google Scholar 

  92. Duchet, P.: Radon and Helly numbers of segment spaces. Ramanujan Math. Soc. Lect. Notes Ser. 5, 57–71 (2008)

    MathSciNet  Google Scholar 

  93. Duchet, P., Meyniel, H.: Convex sets in graphs I. Helly and Radon theorems for graphs and surfaces. European J. Combin. 4(2), 127–132 (1983)

    MathSciNet  MATH  Google Scholar 

  94. Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19, 247–270 (1985)

    MathSciNet  MATH  Google Scholar 

  95. Ekim, T., Erey, A., Heggernes, P., van‘t Hof, P., Mesiter, D.: Computing minimum geodetic sets in proper interval graphs. Lect. Notes Comput. Sci. 7256, 279–290 (2012)

    Google Scholar 

  96. Eroh, L., Oellermann, O.R.: Geodetic and Steiner sets in 3-Steiner distance hereditary graphs. Discrete Math. 308(18), 4212–4220 (2008)

    MathSciNet  MATH  Google Scholar 

  97. Escuadro, H., Gera, R., Hansberg, A., Rad, N.J., Volkmann, L.: Geodetic domination in graphs. J. Combin. Math. Combin. Comput. 77, 89–101 (2011)

    MathSciNet  MATH  Google Scholar 

  98. Everett, M.G., Seidman, S.B.: The hull number of a graph. Discrete Math. 57, 217–223 (1985)

    MathSciNet  MATH  Google Scholar 

  99. Farber, M.: Bridged graphs and geodesic convexity. Discrete Math. 66, 249–257 (1987)

    MathSciNet  MATH  Google Scholar 

  100. Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Alg. Disc. Math. 7(3), 433–444 (1986)

    MathSciNet  MATH  Google Scholar 

  101. Farber, M., Jamison, R.E.: On local convexity in graphs. Discrete Math. 66, 231–247 (1987)

    MathSciNet  MATH  Google Scholar 

  102. Farrugia, A.: Orientable convexity, geodetic and hull numbers in graphs. Discrete Appl. Math. 148(3), 256–262 (2005)

    MathSciNet  MATH  Google Scholar 

  103. Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Math. 4, 322–325 (1970)

    MathSciNet  MATH  Google Scholar 

  104. Gimbel, J.G.: Some remarks on the convexity number of a graph. Graphs Combin. 19, 357–361 (2003)

    MathSciNet  MATH  Google Scholar 

  105. Goddard, W.: A note on Steiner-distance-hereditary graphs. J. Combin. Math. Combin. Comput. 40, 167–170 (2002)

    MathSciNet  MATH  Google Scholar 

  106. Gruber, P.M., Wills, J.M.: Handbook of Convex Geometry, (v. A-B). North-Holland, Amsterdam (1993)

    Google Scholar 

  107. Gutin, G., Yeo, A.: On the number of connected convex subgraphs of a connected acyclic digraph. Discrete Appl. Math. 157(7), 1660–1662 (2009)

    MathSciNet  MATH  Google Scholar 

  108. Hammark, R., Imrich, R., Klavžar, S.: Handbook of Product Graphs. CRC Press, Boca Raton (2011)

    Google Scholar 

  109. Hansberg, A., Volkmann, L.: On the geodetic and geodetic domination numbers of a graph. Discrete Math. 310(15–16), 2140–2146 (2010)

    MathSciNet  MATH  Google Scholar 

  110. Hansen, P., van Omme, N.: On pitfalls in computing the geodetic number of a graph. Optim. Lett. 1(3), 299–307 (2007)

    MathSciNet  MATH  Google Scholar 

  111. Harary, F.: Achievement and avoidance games for graphs. Ann. Discrete Math. 13(11), 111–119 (1982)

    Google Scholar 

  112. Harary, F., Nieminen, J.: Convexity in graphs. J. Differential Geom. 16(2), 185–190 (1981)

    MathSciNet  MATH  Google Scholar 

  113. Harary, F., Loukakis, E., Tsouros, C.: The geodetic number of a graph. Math. Comput. Modelling 17(11), 89–95 (1993)

    MathSciNet  MATH  Google Scholar 

  114. Hasegawa, Y., Saito, A.: Graphs with small boundary. Discrete Math. 307(14), 1801–1807 (2007)

    MathSciNet  MATH  Google Scholar 

  115. Henning, M.A., Nielsen, M.H., Oellermann, O.R.: Local Steiner convexity. European J. Combin. 30, 1186–1193 (2009)

    MathSciNet  MATH  Google Scholar 

  116. Hernando, C., Jiang, T., Mora, M., Pelayo, I.M., Seara, C.: On the Steiner, geodetic and hull numbers of graphs. Discrete Math. 293(1–3), 139–154 (2005)

    MathSciNet  MATH  Google Scholar 

  117. Hernando, C., Mora, M., Pelayo, I.M., Seara, C.: Some structural, metric and convex properties on the boundary of a graph. Electron. Notes Discrete Math. 24, 203–209 (2006)

    MathSciNet  Google Scholar 

  118. Hernando, C., Mora, M., Pelayo, I.M., Seara, C.: Some structural, metric and convex properties of the boundary of a graph. Ars Combin. 109, 267–283 (2013)

    MathSciNet  Google Scholar 

  119. Hernando, C., Mora, M., Pelayo, I.M., Seara, C.: On monophonic sets in graphs. Submitted.

    Google Scholar 

  120. Howorka, E.: A characterization of distance-hereditary graphs. Quart. J. Math. Oxford Ser. 2 28(112), 417–420 (1977)

    Google Scholar 

  121. Howorka, E.: A characterization of Ptolemaic graphs. J. Graph Theory 5(3), 323–331 (1981)

    MathSciNet  MATH  Google Scholar 

  122. Hung, J.-T., Tong, L.-D., Wang, H.-T.: The hull and geodetic numbers of orientations of graphs. Discrete Math. 309(8), 2134–2139 (2009)

    MathSciNet  MATH  Google Scholar 

  123. Imrich, W., Klavžar, S.: A convexity lemma and expansion procedures for bipartite graphs. European J. Combin. 19(6), 677–685 (1998)

    MathSciNet  MATH  Google Scholar 

  124. Isaksen, D.C., Robinson, B.: Triangle-free polyconvex graphs. Ars Combin. 64, 259–263 (2002)

    MathSciNet  MATH  Google Scholar 

  125. Jamil, F.P.; Aniversario, I.S., Canoy, S.R.: On closed and upper closed geodetic numbers of graphs. Ars Combin. 84, 191–203 (2007)

    MathSciNet  MATH  Google Scholar 

  126. Jamil, F.P.; Aniversario, I.S., Canoy, S.R.: The closed geodetic numbers of the corona and composition of graphs. Util. Math. 82, 135–153 (2010)

    MathSciNet  MATH  Google Scholar 

  127. Jamison, R.E.: Copoints in antimatroids. Congr. Numer. 29, 535–544 (1980)

    MathSciNet  Google Scholar 

  128. Jamison, R.E.: Partition numbers for trees and ordered sets. Pacific J. Math. 96, 115–140 (1981)

    MathSciNet  MATH  Google Scholar 

  129. Jamison, R.E., Nowakowsky, R.: A Helly theorem for convexity in graphs. Disc. Math. 51, 35–39 (1984)

    MATH  Google Scholar 

  130. Jiang, T., Pelayo, I.M., Pritikin, D.: Geodesic Convexity and Cartesian Products in Graphs. Manuscript (2004)

    Google Scholar 

  131. Karp, R.M.: Reducibility among combinatorial problems. In Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Google Scholar 

  132. Kay, D.C., Womble, E.W.: Axiomatic convexity theory and relationships between the Carathéodory, Helly and Radon numbers. Pacific J. Math. 38, 471–485 (1971)

    MathSciNet  MATH  Google Scholar 

  133. Klee, V.: What is a convex set. Am. Math. Mon. 78, 616–631 (1971)

    MathSciNet  MATH  Google Scholar 

  134. Korte, B., Lovász, L.: Homomorphisms and Ramsey properties of antimatroids. Discrete Appl. Math. 15 (2–3), 283–290 (1986)

    MathSciNet  MATH  Google Scholar 

  135. Korte, B., Lovász, L., Schrader, R.: Gredoids. Springer, Berlin (1991)

    Google Scholar 

  136. Kubicka, E., Kubicki, G., Oellermann, O.R.: Steiner intervals in graphs. Discrete Appl. Math. 81(1–3), 181–190 (1998)

    MathSciNet  MATH  Google Scholar 

  137. Lemanska, M.: Weakly convex and convex domination numbers. Opuscula Math. 24(2), 181–188 (2004)

    MathSciNet  MATH  Google Scholar 

  138. Levi, F.W.: On Helly’s theorem and the axioms of convexity. J. Indian Math. Soc. 15, 65–76 (1951)

    MATH  Google Scholar 

  139. Lu, C.: The geodetic numbers of graphs and digraphs. Sci. China Ser. A: Math. 50(8), 1163–1172 (2007)

    MATH  Google Scholar 

  140. Malvestuto, F.M., Mezzini, M., Moscarini, M.: Characteristic properties and recognition of graphs in which geodesic and monophonic convexities are equivalent. Discrete Math. Algorithms Appl. 4(4), 1250063, pp. 14 (2012)

    Google Scholar 

  141. McKee, T.A., McMorris, F.R.: Topics in intersection graph theory. SIAM Monographs in Discrete Mathematics and Applications. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  142. Morgana, M.A., Mulder, H.M.: The induced path convexity, betweenness, and svelte graphs. Discrete Math. 254(1–3), 349–370 (2002)

    MathSciNet  MATH  Google Scholar 

  143. Mulder, H.M.: The interval function of a graph. Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam (1980)

    MATH  Google Scholar 

  144. Mulder, H.M., Nebeský, L.: Axiomatic characterization of the interval function of a graph. European J. Combin. 30(5), 1172–1185 (2009)

    MathSciNet  MATH  Google Scholar 

  145. Müller, H., Brandstädt, A.: The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs. Theoret. Comput. Sci. 53(2–3), 257–265 (1987)

    MathSciNet  MATH  Google Scholar 

  146. Muntean, R., Zhang, P.: On geodomination in graphs. Congr. Numer. 143, 161–174 (2000)

    MathSciNet  MATH  Google Scholar 

  147. Muntean, R., Zhang, P.: k-geodomination in graphs. Ars Combin. 63, 33–47 (2002)

    Google Scholar 

  148. Nebeský, L.: A characterization of the interval function of a connected graph. Czechoslovak Math. J. 44(1), 173–178 (1994)

    MathSciNet  MATH  Google Scholar 

  149. Nielsen, M.H., Oellermann, O.R.: Helly theorems for 3-Steiner and 3-monophonic convexity in graphs. Discrete Math. 311(10–11), 872–880 (2011)

    MathSciNet  MATH  Google Scholar 

  150. Nielsen, M.H., Oellermann, O.R.: Steiner trees and convex geometries. SIAM J. Discrete Math. 23(2), 680–693 (2011)

    MathSciNet  Google Scholar 

  151. Oellermann, O.R., Peters-Fransen, J.: The strong metric dimension of graphs and digraphs. Discrete Appl. Math. 155, 356–364 (2007)

    MathSciNet  MATH  Google Scholar 

  152. Oellermann, O.R., Puertas, M.L.: Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs. Discrete Math. 307(1), 88–96 (2007)

    MathSciNet  MATH  Google Scholar 

  153. Parker, D.B., Westhoff, R.F., Wolf, M.J.: Two-path convexity in clone-free regular multipartite tournaments. Australas. J. Combin. 36, 177–196 (2006)

    MathSciNet  MATH  Google Scholar 

  154. Parvathy, K.S., Vijayakumar, A.: Geodesic iteration number. Proceedings of the Conference on Graph Connections, India, 1998, pp. 91–94. Allied Publishers, New Delhi (1999)

    Google Scholar 

  155. Pelayo, I.M.: Comment on “The Steiner number of a graph” by G. Chartrand and P. Zhang [Discrete Math. 242, 41–54 (2002)]. Discrete Math. 280, 259–263 (2004)

    Google Scholar 

  156. Pelayo, I.M.: Generalizing the Krein-Milman property in graph convexity spaces: a short survey. Ramanujan Math. Soc. Lect. Notes Ser. 5, 131–142 (2008)

    MathSciNet  Google Scholar 

  157. Peterin, I.: The pre-hull number and lexicographic product. Discrete Math. 312(14), 2153–2157 (2012)

    MathSciNet  MATH  Google Scholar 

  158. Peterin, I.: Intervals and convex sets in strong product of graphs. Graphs Combin. 29(3), 705–714 (2013)

    MathSciNet  MATH  Google Scholar 

  159. Polat, N.: A Helly theorem for geodesic convexity in strongly dismantlable graphs. Discrete Math. 140, 119–127 (1995)

    MathSciNet  MATH  Google Scholar 

  160. Polat, N.: Graphs without isometric rays and invariant subgraph properties. I. J. Graph Theory 27(2), 99–109 (1998)

    MathSciNet  MATH  Google Scholar 

  161. Polat, N.: On isometric subgraphs of infinite bridged graphs and geodesic convexity. Discrete Math. 244(1–3), 399–416 (2002)

    MathSciNet  MATH  Google Scholar 

  162. Polat, N.: On constructible graphs, locally Helly graphs, and convexity. J. Graph Theory 43(4), 280–298 (2003)

    MathSciNet  MATH  Google Scholar 

  163. Polat, N., Sabidussi, G.: On the geodesic pre-hull number of a graph. European J. Combin. 30(5), 1205–1220 (2009)

    MathSciNet  MATH  Google Scholar 

  164. Raczek, J.: NP-completeness of weakly convex and convex dominating set decision problems. Opuscula Math. 24(2), 189–196 (2004)

    MathSciNet  MATH  Google Scholar 

  165. Sampathkumar, E.: Convex sets in a graph. Indian J. Pure Appl. Math. 15(10), 1065–1071 (1984)

    MathSciNet  MATH  Google Scholar 

  166. Santhakumaran, A.P., John, J.: Edge geodetic number of a graph. J. Discrete Math. Sci. Cryptogr. 10(3), 415–432 (2007)

    MathSciNet  MATH  Google Scholar 

  167. Sierksma, G.: Carathéodory and Helly-numbers of convex product structures. Pacific J. Math. 61, 275–282 (1975)

    MathSciNet  MATH  Google Scholar 

  168. Sierksma, G.: Relationships between Carathéodory, Helly, Radon and exchange numbers of convexity spaces. Nieuw Archief voor Wisk. 25(2), 115–132 (1977)

    MathSciNet  MATH  Google Scholar 

  169. Soltan, V.P.: Metric convexity in graphs. Studia Univ. Babeş-Bolyai Math. 36(4), 3–43 (1991)

    MathSciNet  MATH  Google Scholar 

  170. Soltan, V., Chepoi, V.: Conditions for invariance of set diameter under d-convexification in a graph. Cyber. Syst. Anal. 19(6), 750–756 (1983)

    MathSciNet  MATH  Google Scholar 

  171. Tong, L.-D.: The forcing hull and forcing geodetic numbers of graphs. Discrete Appl. Math. 157(5), 1159–1163 (2009)

    MathSciNet  MATH  Google Scholar 

  172. Tong, L.-D.: Geodetic sets and Steiner sets in graphs. Discrete Math. 309(12), 4205–4207 (2009)

    MathSciNet  MATH  Google Scholar 

  173. Tong, L.-D.: The (a, b)-forcing geodetic graphs. Discrete Math. 309(6), 1623–1628 (2009)

    Google Scholar 

  174. Tong, L.-D., Yen, P.-L., Farrugia, A.: The convexity spectra of graphs. Discrete Appl. Math. 156(10), 1838–1845 (2008)

    MathSciNet  MATH  Google Scholar 

  175. Van de Vel, M.: Theory of Convex Structures. North-Holland, Amsterdam (1993)

    MATH  Google Scholar 

  176. Wang, F.-H.: The lower and upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori. Int. J. Comput. Math. 87(12), 2677–2687 (2010)

    MathSciNet  MATH  Google Scholar 

  177. Wang, F.-H., Wang, Y.-L., Chang, J.-M.: The lower and upper forcing geodetic numbers of block: cactus graphs. Eur. J. Oper. Res. 175(1), 238–245 (2006)

    MathSciNet  MATH  Google Scholar 

  178. Ye, Y., Lu, C., Liu, Q.: The geodetic numbers of cartesian products of graphs. Math. Appl. (Wuhan) 20(1), 158–163 (2007)

    Google Scholar 

  179. Yen, P.-L.: A study of convexity spectra in directed graphs. Doctorate Dissertation, National Sun Yat-sen University, Taywan (2011)

    Google Scholar 

  180. Yero, I.G., Rodríguez-Velázquez, J.A.: Analogies between the geodetic number and the Steiner number of some classes of graphs. submitted

    Google Scholar 

  181. Zhang, P.: The upper forcing geodetic number of a graph. Ars Combin. 62, 3–15 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Agricultural Engineering of Barcelona, Barcelona, Spain

    Ignacio M. Pelayo

Authors
  1. Ignacio M. Pelayo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Ignacio M. Pelayo

About this chapter

Cite this chapter

Pelayo, I.M. (2013). Computational Complexity. In: Geodesic Convexity in Graphs. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8699-2_7

Download citation

Publish with us

Policies and ethics

Search

Navigation