On graphs with a unique minimum hull set

Gary Chartrand; Ping Zhang

Discussiones Mathematicae Graph Theory (2001)

  • Volume: 21, Issue: 1, page 31-42
  • ISSN: 2083-5892

Abstract

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We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link L ( v i ) = G i for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.

How to cite

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Gary Chartrand, and Ping Zhang. "On graphs with a unique minimum hull set." Discussiones Mathematicae Graph Theory 21.1 (2001): 31-42. <http://eudml.org/doc/270467>.

@article{GaryChartrand2001,
abstract = {We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L(v_i) = G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.},
author = {Gary Chartrand, Ping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {geodetic set; geodetic number; convex hull; hull set; hull number; hull graph},
language = {eng},
number = {1},
pages = {31-42},
title = {On graphs with a unique minimum hull set},
url = {http://eudml.org/doc/270467},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Gary Chartrand
AU - Ping Zhang
TI - On graphs with a unique minimum hull set
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 1
SP - 31
EP - 42
AB - We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L(v_i) = G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.
LA - eng
KW - geodetic set; geodetic number; convex hull; hull set; hull number; hull graph
UR - http://eudml.org/doc/270467
ER -

References

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  1. [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). Zbl0688.05017
  2. [2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, to appear. Zbl0987.05047
  3. [3] G. Chartrand, F. Harary and P. Zhang, On the hull number of a graph, Ars Combin. 57 (2000) 129-138. Zbl1064.05049
  4. [4] G. Chartrand and P. Zhang, The geodetic number of an oriented graph, European J. Combin. 21 (2) (2000) 181-189, doi: 10.1006/eujc.1999.0301. Zbl0941.05033
  5. [5] G. Chartrand and P. Zhang, The forcing geodetic number of a graph, Discuss. Math. Graph Theory 19 (1999) 45-58, doi: 10.7151/dmgt.1084. Zbl0927.05025
  6. [6] G. Chartrand and P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. to appear. Zbl1006.05022
  7. [7] M.G. Everett and S.B. Seidman, The hull number of a graph, Discrete Math. 57 (1985) 217-223, doi: 10.1016/0012-365X(85)90174-8. Zbl0584.05044
  8. [8] F. Harary and J. Nieminen, Convexity in graphs, J. Differential Geom. 16 (1981) 185-190. Zbl0493.05037
  9. [9] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Mathl. Comput. Modelling. 17 (11) (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2. Zbl0825.68490
  10. [10] H.M. Mulder, The Interval Function of a Graph (Methematisch Centrum, Amsterdam, 1980). 
  11. [11] H.M. Mulder, The expansion procedure for graphs, in: Contemporary Methods in Graph Theory ed., R. Bodendiek (Wissenschaftsverlag, Mannheim, 1990) 459-477. Zbl0744.05064
  12. [12] L. Nebeský, A characterization of the interval function of a connected graph, Czech. Math. J. 44 (119) (1994) 173-178. Zbl0808.05046
  13. [13] L. Nebeský, Characterizing of the interval function of a connected graph, Math. Bohem. 123 (1998) 137-144. Zbl0937.05036

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