Discussiones Mathematicae Graph Theory 32(4) (2012) 685-704
doi: 10.7151/dmgt.1638

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Minimal Trees and Monophonic Convexity

Jose Cáceresa, Ortrud R. Oellermannb
M.L. Puertasa

a Department of Statistics and Applied Mathematics
University of Almeria, 04120, Almeria, Spain
b Department of Mathematics and Statistics., University of Winnipeg,
515 Portage Ave, Winnipeg, R3B 2E9, Canada


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.

Keywords: minimal trees, monophonic intervals of sets, k-monophonic convexity, convex geometries

2010 Mathematics Subject Classification: 05C75, 05C12, 05C17, 05C05.


[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. Brandst{ä}dt, 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.
[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).

Received 11 July 2011
Revised 20 December 2011
Accepted 21 December 2011