Boundary Sets

  • Ignacio M. Pelayo
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Let S be a set of vertices of a connected graph G = (V, E). The eccentricity e S (v) of a vertex vS is the maximum distance between v and any other vertex of S, i.e., e S (v) = max{d(v, u): uS}. A vertex uS is said to be a contour vertex of S if \(e_{S}(u) \geq e_{S}(v)\) for every neighbor v of u in S. The set of all contour vertices of S is called the contour set of S and is denoted by Ct(S).

References

  1. 1.
    Anand, B.S., Changat, M., Klavžar, S., Peterin, I.: Convex sets in lexicographic products of graphs. Graphs Combin. 28, 77–84 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Aniversario, I.S., Jamil, F.P., Canoy, S.R.: The closed geodetic numbers of graphs. Util. Math. 74, 3–18 (2007)MathSciNetMATHGoogle Scholar
  3. 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)MathSciNetMATHGoogle Scholar
  4. 4.
    Artigas, D., Dantas, S., Dourado, M.C., Szwarcfiter, J.L.: Partitioning a graph into convex sets. Discrete Math. 311(17), 1968–1977 (2011)MathSciNetMATHGoogle Scholar
  5. 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)MathSciNetGoogle Scholar
  6. 6.
    Atici, M.: Computational complexity of geodetic set. Int. J. Comput. Math. 79(5), 587–591 (2002)MathSciNetMATHGoogle Scholar
  7. 7.
    Atici, M.: On the edge geodetic number of a graph. Int. J. Comput. Math. 80(7), 853–861 (2003)MathSciNetMATHGoogle Scholar
  8. 8.
    Atici, M., Vince, A.: Geodesics in graphs, an extremal set problem and perfect Hash families. Graphs Combin. 18(3), 403–413 (2002)MathSciNetMATHGoogle Scholar
  9. 9.
    Bandelt, H.-J., Chepoi, V.: A Helly theorem in weakly modular spaces. Discrete Math. 125, 25–39 (1996)MathSciNetGoogle Scholar
  10. 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. 11.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41(2), 182–208 (1986)MathSciNetMATHGoogle Scholar
  12. 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)MathSciNetMATHGoogle Scholar
  13. 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. 14.
    Brandstädt, A., V.B., Spinrad, J.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)Google Scholar
  15. 15.
    Brešar, B., Gologranc, T.: On a local 3-Steiner convexity. European J. Combin. 32(8), 1222–1235 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Brešar, B., Tepeh Horvat, A.: On the geodetic number of median graphs. Discrete Math. 308(18), 4044–4051 (2008)MathSciNetMATHGoogle Scholar
  17. 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)MathSciNetMATHGoogle Scholar
  18. 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)MathSciNetMATHGoogle Scholar
  19. 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. 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)MathSciNetMATHGoogle Scholar
  21. 21.
    Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Redwood City (1990)MATHGoogle Scholar
  22. 22.
    Buckley, F., Harary, F., Quintas, L.V.: Extremal results on the geodetic number of a graph. Scientia 2A, 17–26 (1988)MathSciNetGoogle Scholar
  23. 23.
    Cáceres, J., Márquez, A., Oellermann, O.R., Puertas, M.L.: Rebuilding convex sets in graphs. Discrete Math. 297(1–3), 26–37 (2005)MathSciNetMATHGoogle Scholar
  24. 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)MathSciNetMATHGoogle Scholar
  25. 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)MathSciNetMATHGoogle Scholar
  26. 26.
    Cáceres, J., Márquez, A., Puertas, M.L.: Steiner distance and convexity in graphs. European J. Combin. 29(3), 726–736 (2008)MathSciNetMATHGoogle Scholar
  27. 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)MathSciNetMATHGoogle Scholar
  28. 28.
    Cáceres, J., Oellermann, O.R., Puertas, M.L.: Minimal trees and monophonic convexity. Discuss. Math. Graph Theory 32(4), 685–704 (2012)MathSciNetGoogle Scholar
  29. 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)MathSciNetMATHGoogle Scholar
  30. 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)MathSciNetMATHGoogle Scholar
  31. 31.
    Cagaanan, G.B., Canoy, S.R.: Bounds for the geodetic number of the Cartesian product of graphs. Util. Math. 79, 91–98 (2009)MathSciNetMATHGoogle Scholar
  32. 32.
    Calder, J.R.: Some elementary properties of interval convexities. J. London Math. Soc. 3(2), 422–428 (1971)MathSciNetMATHGoogle Scholar
  33. 33.
    Canoy, S.R., Cagaanan, G.B.: On the geodesic and hull numbers of the sum of graphs. Congr. Numer. 161, 97–104 (2003)MathSciNetMATHGoogle Scholar
  34. 34.
    Canoy, S.R., Cagaanan, G.B.: On the hull number of the composition of graphs. Ars Combin. 75, 113–119 (2005)MathSciNetMATHGoogle Scholar
  35. 35.
    Canoy, S.R., Garces, I.J.L.: Convex sets under some graph operations. Graphs Combin. 18(4), 787–793 (2002)MathSciNetMATHGoogle Scholar
  36. 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)MathSciNetMATHGoogle Scholar
  37. 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)MathSciNetGoogle Scholar
  38. 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)MathSciNetMATHGoogle Scholar
  39. 39.
    Chang, G.J., Tong, L.-D., Wang, H.-T.: Geodetic spectra of graphs. European J. Combin. 25(3), 383–391 (2004)MathSciNetMATHGoogle Scholar
  40. 40.
    Changat, M., Mathew, J.: On triangle path convexity in graphs. Discrete Math. 206, 91–95 (1999)MathSciNetMATHGoogle Scholar
  41. 41.
    Changat, M., Mathew, J.: Induced path transit function, monotone and Peano axioms. Discrete Math. 286(3), 185–194 (2004)MathSciNetMATHGoogle Scholar
  42. 42.
    Changat, M., Klavžar, S., Mulder, H.M.: The all-paths transit function of a graph. Czechoslovak Math. J. 51(2), 439–448 (2001)MathSciNetMATHGoogle Scholar
  43. 43.
    Changat, M., Mulder, H.M., Sierksma, G.: Convexities related to path properties on graphs. Discrete Math. 290(2–3), 117–131 (2005)MathSciNetMATHGoogle Scholar
  44. 44.
    Changat, M., Mathew, J., Mulder, H.M.: The induced path function, monotonicity and betweenness. Discrete Appl. Math. 158(5), 426–433 (2010)MathSciNetMATHGoogle Scholar
  45. 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)MathSciNetMATHGoogle Scholar
  46. 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)MathSciNetMATHGoogle Scholar
  47. 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)MathSciNetMATHGoogle Scholar
  48. 48.
    Chartrand, G., Zhang, P.: Convex sets in graphs. Congr. Numer. 136, 19–32 (1999)MathSciNetMATHGoogle Scholar
  49. 49.
    Chartrand, G., Zhang, P.: The forcing geodetic number of a graph. Discuss. Math. Graph Theory 19, 45–58 (1999)MathSciNetMATHGoogle Scholar
  50. 50.
    Chartrand, G., Zhang, P.: The geodetic number of an oriented graph. Europ. J. Combin. 21, 181–189 (2000)MathSciNetMATHGoogle Scholar
  51. 51.
    Chartrand, G., Zhang, P.: On graphs with a unique minimum hull set. Discuss. Math. Graph Theory 21(1), 31–42 (2001)MathSciNetMATHGoogle Scholar
  52. 52.
    Chartrand, G., Zhang, P.: The forcing convexity number of a graph. Czechoslovak Math. J. 51(4), 847–858 (2001)MathSciNetMATHGoogle Scholar
  53. 53.
    Chartrand, G., Zhang, P.: The forcing hull number of a graph. J. Combin. Math. Combin. Comput. 36, 81–94 (2001)MathSciNetMATHGoogle Scholar
  54. 54.
    Chartrand, G., Zhang, P.: Extreme Geodesic Graphs. Czechoslovak Math. J. 52(4), 771–780 (2002)MathSciNetMATHGoogle Scholar
  55. 55.
    Chartrand, G., Zhang, P.: The Steiner number of a graph. Discrete Math. 242, 41–54 (2002)MathSciNetMATHGoogle Scholar
  56. 56.
    Chartrand, G., Oellermann, O.R., Tian, S., Zou, H.: Steiner distance in graphs. Časopis Pěst. Mat. 114(4), 399–410 (1989)MathSciNetMATHGoogle Scholar
  57. 57.
    Chartrand, G., Zhang, P., Harary, F.: Extremal problems in geodetic graph theory. Cong. Numer. 131, 55–66 (1998)MathSciNetMATHGoogle Scholar
  58. 58.
    Chartrand, G., Harary, F., Zhang, P.: Geodetic sets in graphs. Discuss. Math. Graph Theory 20, 129–138 (2000)MathSciNetMATHGoogle Scholar
  59. 59.
    Chartrand, G., Harary, F., Zhang, P.: On the hull number of a graph. Ars Combin. 57, 129–138 (2000)MathSciNetMATHGoogle Scholar
  60. 60.
    Chartrand, G., Harary, F., Swart, H.C., Zhang, P.: Geodomination in graphs. Bull. Inst. Combin. Appl. 31, 51–59 (2001)MathSciNetMATHGoogle Scholar
  61. 61.
    Chartrand, G., Chichisan, A., Wall, C.E., Zhang, P.: On convexity in graphs. Congr. Numer. 148, 33–41 (2001)MathSciNetMATHGoogle Scholar
  62. 62.
    Chartrand, G., Fink, J.F., Zhang, P.: Convexity in oriented graphs. Discrete Math. 116(1–2), 115–126 (2002)MathSciNetMATHGoogle Scholar
  63. 63.
    Chartrand, G., Harary, F., Zhang, P.: On the geodetic number of a graph. Networks 39, 1–6 (2002)MathSciNetMATHGoogle Scholar
  64. 64.
    Chartrand, G., Wall, C.E., Zhang, P.: The convexity number of a graph. Graphs and Combin. 18(2), 209–217 (2002)MathSciNetMATHGoogle Scholar
  65. 65.
    Chartrand, G., Palmer, E.M., Zhang, P.: The geodetic number of a graph: a survey. Congr. Numer. 156, 37–58 (2002)MathSciNetMATHGoogle Scholar
  66. 66.
    Chartrand, G., Erwin, D., Johns, G.L., Zhang, P.: Boundary vertices in graphs. Discrete Math. 263, 25–34 (2003)MathSciNetMATHGoogle Scholar
  67. 67.
    Chartrand, G., Fink, J.F., Zhang, P.: The hull number of an oriented graph. Int. J. Math. Math. Sci. 36, 2265–2275 (2003)MathSciNetGoogle Scholar
  68. 68.
    Chartrand, G., Garry, L., Zhang, P.: The detour number of a graph. Util. Math. 64, 97–113 (2003)MathSciNetMATHGoogle Scholar
  69. 69.
    Chartrand, G., Garry, L., Zhang, P.: On the detour number and geodetic number of a graph. Ars Combin. 72, 3–15 (2004)MathSciNetMATHGoogle Scholar
  70. 70.
    Chartrand, G., Escuadro, H., Zhang, P.: Detour distance in graphs. J. Combin. Math. Combin. Comput. 52, 75–94 (2005)MathSciNetGoogle Scholar
  71. 71.
    Chartrand, G., Lesniak, L., Zhang, P.: Graphs and Digraphs, 5th edn. CRC Press, Boca Raton (2011)Google Scholar
  72. 72.
    Chastand, M., Polat, N.: On geodesic structures of weakly median graphs I. Decomposition and octahedral graphs. Discrete Math. 306(13), 1272–1284 (2006)MathSciNetMATHGoogle Scholar
  73. 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)MathSciNetMATHGoogle Scholar
  74. 74.
    Chepoi, V.: Isometric subgraphs of Hamming graphs and d-convexity. Cyber. Syst. Anal. 24(1), 6–11 (1988)MathSciNetGoogle Scholar
  75. 75.
    Chepoi, V.: Separation of two convex sets in convexity structures. J. Geom. 50(1–2), 30–51 (1994)MathSciNetMATHGoogle Scholar
  76. 76.
    Chepoi, V.: Peakless functions on graphs. Discrete Appl. Math. 73(2), 175–189 (1997)MathSciNetMATHGoogle Scholar
  77. 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)MathSciNetMATHGoogle Scholar
  78. 78.
    Dirac, G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)MathSciNetMATHGoogle Scholar
  79. 79.
    Dong, L., Lu, C., Wang, X.: The upper and lower geodetic numbers of graphs. Ars Combin. 91, 401–409 (2009)MathSciNetMATHGoogle Scholar
  80. 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)MathSciNetGoogle Scholar
  81. 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)MathSciNetMATHGoogle Scholar
  82. 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. 83.
    Dourado, M.C., Protti, F., Szwarcfiter, J.L.: Complexity results related to monophonic convexity. Discrete Appl. Math. 158(12), 1268–1274 (2010)MathSciNetMATHGoogle Scholar
  84. 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)MathSciNetMATHGoogle Scholar
  85. 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)MathSciNetMATHGoogle Scholar
  86. 86.
    Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: On the convexity number of graphs. Graphs Combin. 28(3), 333–345 (2012)MathSciNetMATHGoogle Scholar
  87. 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)MathSciNetMATHGoogle Scholar
  88. 88.
    Dragan, F.F., Nicolai, F., Brandstädt, A.: Convexity and HHD-free graphs. SIAM J. Discrete Math. 12, 119–135 (1999)MathSciNetMATHGoogle Scholar
  89. 89.
    Duchet, P.: Convex sets in graphs II. Minimal path convexity. J. Comb. Theory Ser. B 44(3), 307–316 (1988)MathSciNetMATHGoogle Scholar
  90. 90.
    Duchet, P.: Convexity in combinatorial structures. Rend. Circ. Mat. Palermo 14(2), 261–293 (1987)Google Scholar
  91. 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. 92.
    Duchet, P.: Radon and Helly numbers of segment spaces. Ramanujan Math. Soc. Lect. Notes Ser. 5, 57–71 (2008)MathSciNetGoogle Scholar
  93. 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)MathSciNetMATHGoogle Scholar
  94. 94.
    Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19, 247–270 (1985)MathSciNetMATHGoogle Scholar
  95. 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. 96.
    Eroh, L., Oellermann, O.R.: Geodetic and Steiner sets in 3-Steiner distance hereditary graphs. Discrete Math. 308(18), 4212–4220 (2008)MathSciNetMATHGoogle Scholar
  97. 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)MathSciNetMATHGoogle Scholar
  98. 98.
    Everett, M.G., Seidman, S.B.: The hull number of a graph. Discrete Math. 57, 217–223 (1985)MathSciNetMATHGoogle Scholar
  99. 99.
    Farber, M.: Bridged graphs and geodesic convexity. Discrete Math. 66, 249–257 (1987)MathSciNetMATHGoogle Scholar
  100. 100.
    Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Alg. Disc. Math. 7(3), 433–444 (1986)MathSciNetMATHGoogle Scholar
  101. 101.
    Farber, M., Jamison, R.E.: On local convexity in graphs. Discrete Math. 66, 231–247 (1987)MathSciNetMATHGoogle Scholar
  102. 102.
    Farrugia, A.: Orientable convexity, geodetic and hull numbers in graphs. Discrete Appl. Math. 148(3), 256–262 (2005)MathSciNetMATHGoogle Scholar
  103. 103.
    Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Math. 4, 322–325 (1970)MathSciNetMATHGoogle Scholar
  104. 104.
    Gimbel, J.G.: Some remarks on the convexity number of a graph. Graphs Combin. 19, 357–361 (2003)MathSciNetMATHGoogle Scholar
  105. 105.
    Goddard, W.: A note on Steiner-distance-hereditary graphs. J. Combin. Math. Combin. Comput. 40, 167–170 (2002)MathSciNetMATHGoogle Scholar
  106. 106.
    Gruber, P.M., Wills, J.M.: Handbook of Convex Geometry, (v. A-B). North-Holland, Amsterdam (1993)Google Scholar
  107. 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)MathSciNetMATHGoogle Scholar
  108. 108.
    Hammark, R., Imrich, R., Klavžar, S.: Handbook of Product Graphs. CRC Press, Boca Raton (2011)Google Scholar
  109. 109.
    Hansberg, A., Volkmann, L.: On the geodetic and geodetic domination numbers of a graph. Discrete Math. 310(15–16), 2140–2146 (2010)MathSciNetMATHGoogle Scholar
  110. 110.
    Hansen, P., van Omme, N.: On pitfalls in computing the geodetic number of a graph. Optim. Lett. 1(3), 299–307 (2007)MathSciNetMATHGoogle Scholar
  111. 111.
    Harary, F.: Achievement and avoidance games for graphs. Ann. Discrete Math. 13(11), 111–119 (1982)Google Scholar
  112. 112.
    Harary, F., Nieminen, J.: Convexity in graphs. J. Differential Geom. 16(2), 185–190 (1981)MathSciNetMATHGoogle Scholar
  113. 113.
    Harary, F., Loukakis, E., Tsouros, C.: The geodetic number of a graph. Math. Comput. Modelling 17(11), 89–95 (1993)MathSciNetMATHGoogle Scholar
  114. 114.
    Hasegawa, Y., Saito, A.: Graphs with small boundary. Discrete Math. 307(14), 1801–1807 (2007)MathSciNetMATHGoogle Scholar
  115. 115.
    Henning, M.A., Nielsen, M.H., Oellermann, O.R.: Local Steiner convexity. European J. Combin. 30, 1186–1193 (2009)MathSciNetMATHGoogle Scholar
  116. 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)MathSciNetMATHGoogle Scholar
  117. 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)MathSciNetGoogle Scholar
  118. 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)MathSciNetGoogle Scholar
  119. 119.
    Hernando, C., Mora, M., Pelayo, I.M., Seara, C.: On monophonic sets in graphs. Submitted.Google Scholar
  120. 120.
    Howorka, E.: A characterization of distance-hereditary graphs. Quart. J. Math. Oxford Ser. 2 28(112), 417–420 (1977)Google Scholar
  121. 121.
    Howorka, E.: A characterization of Ptolemaic graphs. J. Graph Theory 5(3), 323–331 (1981)MathSciNetMATHGoogle Scholar
  122. 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)MathSciNetMATHGoogle Scholar
  123. 123.
    Imrich, W., Klavžar, S.: A convexity lemma and expansion procedures for bipartite graphs. European J. Combin. 19(6), 677–685 (1998)MathSciNetMATHGoogle Scholar
  124. 124.
    Isaksen, D.C., Robinson, B.: Triangle-free polyconvex graphs. Ars Combin. 64, 259–263 (2002)MathSciNetMATHGoogle Scholar
  125. 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)MathSciNetMATHGoogle Scholar
  126. 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)MathSciNetMATHGoogle Scholar
  127. 127.
    Jamison, R.E.: Copoints in antimatroids. Congr. Numer. 29, 535–544 (1980)MathSciNetGoogle Scholar
  128. 128.
    Jamison, R.E.: Partition numbers for trees and ordered sets. Pacific J. Math. 96, 115–140 (1981)MathSciNetMATHGoogle Scholar
  129. 129.
    Jamison, R.E., Nowakowsky, R.: A Helly theorem for convexity in graphs. Disc. Math. 51, 35–39 (1984)MATHGoogle Scholar
  130. 130.
    Jiang, T., Pelayo, I.M., Pritikin, D.: Geodesic Convexity and Cartesian Products in Graphs. Manuscript (2004)Google Scholar
  131. 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. 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)MathSciNetMATHGoogle Scholar
  133. 133.
    Klee, V.: What is a convex set. Am. Math. Mon. 78, 616–631 (1971)MathSciNetMATHGoogle Scholar
  134. 134.
    Korte, B., Lovász, L.: Homomorphisms and Ramsey properties of antimatroids. Discrete Appl. Math. 15 (2–3), 283–290 (1986)MathSciNetMATHGoogle Scholar
  135. 135.
    Korte, B., Lovász, L., Schrader, R.: Gredoids. Springer, Berlin (1991)Google Scholar
  136. 136.
    Kubicka, E., Kubicki, G., Oellermann, O.R.: Steiner intervals in graphs. Discrete Appl. Math. 81(1–3), 181–190 (1998)MathSciNetMATHGoogle Scholar
  137. 137.
    Lemanska, M.: Weakly convex and convex domination numbers. Opuscula Math. 24(2), 181–188 (2004)MathSciNetMATHGoogle Scholar
  138. 138.
    Levi, F.W.: On Helly’s theorem and the axioms of convexity. J. Indian Math. Soc. 15, 65–76 (1951)MATHGoogle Scholar
  139. 139.
    Lu, C.: The geodetic numbers of graphs and digraphs. Sci. China Ser. A: Math. 50(8), 1163–1172 (2007)MATHGoogle Scholar
  140. 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. 141.
    McKee, T.A., McMorris, F.R.: Topics in intersection graph theory. SIAM Monographs in Discrete Mathematics and Applications. SIAM, Philadelphia (1999)MATHGoogle Scholar
  142. 142.
    Morgana, M.A., Mulder, H.M.: The induced path convexity, betweenness, and svelte graphs. Discrete Math. 254(1–3), 349–370 (2002)MathSciNetMATHGoogle Scholar
  143. 143.
    Mulder, H.M.: The interval function of a graph. Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam (1980)MATHGoogle Scholar
  144. 144.
    Mulder, H.M., Nebeský, L.: Axiomatic characterization of the interval function of a graph. European J. Combin. 30(5), 1172–1185 (2009)MathSciNetMATHGoogle Scholar
  145. 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)MathSciNetMATHGoogle Scholar
  146. 146.
    Muntean, R., Zhang, P.: On geodomination in graphs. Congr. Numer. 143, 161–174 (2000)MathSciNetMATHGoogle Scholar
  147. 147.
    Muntean, R., Zhang, P.: k-geodomination in graphs. Ars Combin. 63, 33–47 (2002)Google Scholar
  148. 148.
    Nebeský, L.: A characterization of the interval function of a connected graph. Czechoslovak Math. J. 44(1), 173–178 (1994)MathSciNetMATHGoogle Scholar
  149. 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)MathSciNetMATHGoogle Scholar
  150. 150.
    Nielsen, M.H., Oellermann, O.R.: Steiner trees and convex geometries. SIAM J. Discrete Math. 23(2), 680–693 (2011)MathSciNetGoogle Scholar
  151. 151.
    Oellermann, O.R., Peters-Fransen, J.: The strong metric dimension of graphs and digraphs. Discrete Appl. Math. 155, 356–364 (2007)MathSciNetMATHGoogle Scholar
  152. 152.
    Oellermann, O.R., Puertas, M.L.: Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs. Discrete Math. 307(1), 88–96 (2007)MathSciNetMATHGoogle Scholar
  153. 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)MathSciNetMATHGoogle Scholar
  154. 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. 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. 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)MathSciNetGoogle Scholar
  157. 157.
    Peterin, I.: The pre-hull number and lexicographic product. Discrete Math. 312(14), 2153–2157 (2012)MathSciNetMATHGoogle Scholar
  158. 158.
    Peterin, I.: Intervals and convex sets in strong product of graphs. Graphs Combin. 29(3), 705–714 (2013)MathSciNetMATHGoogle Scholar
  159. 159.
    Polat, N.: A Helly theorem for geodesic convexity in strongly dismantlable graphs. Discrete Math. 140, 119–127 (1995)MathSciNetMATHGoogle Scholar
  160. 160.
    Polat, N.: Graphs without isometric rays and invariant subgraph properties. I. J. Graph Theory 27(2), 99–109 (1998)MathSciNetMATHGoogle Scholar
  161. 161.
    Polat, N.: On isometric subgraphs of infinite bridged graphs and geodesic convexity. Discrete Math. 244(1–3), 399–416 (2002)MathSciNetMATHGoogle Scholar
  162. 162.
    Polat, N.: On constructible graphs, locally Helly graphs, and convexity. J. Graph Theory 43(4), 280–298 (2003)MathSciNetMATHGoogle Scholar
  163. 163.
    Polat, N., Sabidussi, G.: On the geodesic pre-hull number of a graph. European J. Combin. 30(5), 1205–1220 (2009)MathSciNetMATHGoogle Scholar
  164. 164.
    Raczek, J.: NP-completeness of weakly convex and convex dominating set decision problems. Opuscula Math. 24(2), 189–196 (2004)MathSciNetMATHGoogle Scholar
  165. 165.
    Sampathkumar, E.: Convex sets in a graph. Indian J. Pure Appl. Math. 15(10), 1065–1071 (1984)MathSciNetMATHGoogle Scholar
  166. 166.
    Santhakumaran, A.P., John, J.: Edge geodetic number of a graph. J. Discrete Math. Sci. Cryptogr. 10(3), 415–432 (2007)MathSciNetMATHGoogle Scholar
  167. 167.
    Sierksma, G.: Carathéodory and Helly-numbers of convex product structures. Pacific J. Math. 61, 275–282 (1975)MathSciNetMATHGoogle Scholar
  168. 168.
    Sierksma, G.: Relationships between Carathéodory, Helly, Radon and exchange numbers of convexity spaces. Nieuw Archief voor Wisk. 25(2), 115–132 (1977)MathSciNetMATHGoogle Scholar
  169. 169.
    Soltan, V.P.: Metric convexity in graphs. Studia Univ. Babeş-Bolyai Math. 36(4), 3–43 (1991)MathSciNetMATHGoogle Scholar
  170. 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)MathSciNetMATHGoogle Scholar
  171. 171.
    Tong, L.-D.: The forcing hull and forcing geodetic numbers of graphs. Discrete Appl. Math. 157(5), 1159–1163 (2009)MathSciNetMATHGoogle Scholar
  172. 172.
    Tong, L.-D.: Geodetic sets and Steiner sets in graphs. Discrete Math. 309(12), 4205–4207 (2009)MathSciNetMATHGoogle Scholar
  173. 173.
    Tong, L.-D.: The (a, b)-forcing geodetic graphs. Discrete Math. 309(6), 1623–1628 (2009)Google Scholar
  174. 174.
    Tong, L.-D., Yen, P.-L., Farrugia, A.: The convexity spectra of graphs. Discrete Appl. Math. 156(10), 1838–1845 (2008)MathSciNetMATHGoogle Scholar
  175. 175.
    Van de Vel, M.: Theory of Convex Structures. North-Holland, Amsterdam (1993)MATHGoogle Scholar
  176. 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)MathSciNetMATHGoogle Scholar
  177. 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)MathSciNetMATHGoogle Scholar
  178. 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. 179.
    Yen, P.-L.: A study of convexity spectra in directed graphs. Doctorate Dissertation, National Sun Yat-sen University, Taywan (2011)Google Scholar
  180. 180.
    Yero, I.G., Rodríguez-Velázquez, J.A.: Analogies between the geodetic number and the Steiner number of some classes of graphs. submittedGoogle Scholar
  181. 181.
    Zhang, P.: The upper forcing geodetic number of a graph. Ars Combin. 62, 3–15 (2002)MathSciNetMATHGoogle Scholar

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© Ignacio M. Pelayo 2013

Authors and Affiliations

  • Ignacio M. Pelayo
    • 1
  1. 1.School of Agricultural Engineering of BarcelonaBarcelonaSpain

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