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2012 | 32 | 4 | 685-704
Tytuł artykułu

Minimal trees and monophonic convexity

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.
Wydawca
Rocznik
Tom
32
Numer
4
Strony
685-704
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-07-11
poprawiono
2011-12-20
zaakceptowano
2011-12-21
Twórcy
  • Department of Statistics and Applied Mathematics, University of Almeria, 04120, Almeria, Spain
  • Department of Mathematics and Statistics., University of Winnipeg, 515 Portage Ave, Winnipeg, R3B 2E9, Canada
  • Department of Statistics and Applied Mathematics, University of Almeria, 04120, Almeria, Spain
Bibliografia
  • [1] M. Atici, Computational complexity of geodetic set, Int. J. Comput. Math. 79 (2002) 587-591, doi: 10.1080/00207160210954.
  • [2] H.-J. Bandelt and H.M. Mulder, Distance-hereditary graphs, J. Combin. Theory (B) 41 (1986) 182-208, doi: 10.1016/0095-8956(86)90043-2.
  • [3] J.M. Bilbao and P.H. Edelman, The Shapley value on convex geometries, Discrete Appl. Math 103 (2000) 33-40, doi: 10.1016/S0166-218X(99)00218-8.
  • [4] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A survey (SIAM Monogr. Discrete Math. Appl., Philidelphia, 1999).
  • [5] J. Cáceres and O.R. Oellermann, On 3-Steiner simplicial orderings, Discrete Math. 309 (2009) 5825-5833, doi: 10.1016/j.disc.2008.05.047.
  • [6] J. Cáceres, O.R. Oellermann and M.L. Puertas, m₃³-convex geometries are A-free, arXiv.org 1107.1048, (2011) 1-15.
  • [7] M. Changat and J. Mathew, Induced path transit function, monotone and Peano axioms, Discrete Math. 286 (2004) 185-194, doi: 10.1016/j.disc.2004.02.017.
  • [8] M. Changat, J. Mathew and H.M. Mulder, The induced path function, monotonicity and betweenness, Discrete Appl. Math. 158 (2010) 426-433, doi: 10.1016/j.dam.2009.10.004.
  • [9] G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs: Fifth Edition (Chapman and Hall, New York, 1996).
  • [10] F.F. Dragan, F. Nicolai and A. Brandstädt, Convexity and HHD-free graphs, SIAM J. Discrete Math. 12 (1999) 119-135, doi: 10.1137/S0895480195321718.
  • [11] M. Farber and R.E. Jamison, Convexity in graphs and hypergraphs, SIAM J. Alg. Discrete Meth. 7 (1986) 433-444, doi: 10.1137/0607049.
  • [12] E. Howorka, A characterization of distance hereditary graphs, Quart. J. Math. Oxford 28 (1977) 417-420, doi: 10.1093/qmath/28.4.417.
  • [13] B. Jamison and S. Olariu, On the semi-perfect elimination, Adv. in Appl. Math. 9 (1988) 364-376, doi: 10.1016/0196-8858(88)90019-X.
  • [14] E. Kubicka, G. Kubicki and O.R. Oellermann, Steiner intervals in graphs, Discrete Math. 81 (1998) 181-190, doi: 10.1016/S0166-218X(97)00084-X.
  • [15] M. Nielsen and O.R. Oellermann, Steiner trees and convex geometries, SIAM J. Discrete Math. 23 (2009) 680-693, doi: 10.1137/070691383.
  • [16] O.R. Oellermann, Convexity notions in graphs (2006) 1-4. http://www-ma2.upc.edu/seara/wmcgt06/.
  • [17] O.R. Oellermann and M.L. Puertas, Steiner intervals and Steiner geodetic numbers in distance hereditary graphs, Discrete Math. 307 (2007) 88-96, doi: 10.1016/j.disc.2006.04.037.
  • [18] M.J.L. Van de Vel, Theory of convex structures (North-Holland, Amsterdam, 1993).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1638
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