Taiwanese Journal of Mathematics


Tanja Gologranc

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Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set $W$ on $k$ vertices in a connected graph $G$ is a tree with the smallest number of edges in $G$ that contains all vertices of $W$. The Steiner interval $I(W)$ of $W$ consists of all vertices in $G$ that lie on some Steiner tree with respect to $W$. Moreover, a set $S$ of vertices in a graph $G$ is $k$-Steiner convex, denoted $g_k$-convex, if the Steiner interval $I(W)$ of every set $W$ on $k$ vertices is contained in $S$. In this paper we consider two types of local convexities. In particular, for every $k \gt 3$, we characterize graphs with $g_k$-convex closed neighborhoods around all vertices of the graph. Then we follow with a characterization of graphs with $g_4$-convex closed neighborhoods around all $g_4$-convex sets of the graph.

Article information

Taiwanese J. Math., Volume 19, Number 5 (2015), 1325-1340.

First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 05C12: Distance in graphs 05C75: Structural characterization of families of graphs

Steiner interval distance Steiner convexity local convexities


Gologranc, Tanja. GRAPHS WITH 4-STEINER CONVEX BALLS. Taiwanese J. Math. 19 (2015), no. 5, 1325--1340. doi:10.11650/tjm.19.2015.4403. https://projecteuclid.org/euclid.twjm/1499133711

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  • B. Bre\ptmrs šar, M. Changat, J. Mathews, I. Peterin, P. G. Narasimha-Shenoi and A. Tepeh Horvat, Steiner intervals, geodesic intervals, and betweenness, Discrete Math., 309 (2009), 6114-6125.
  • B. Bre\ptmrs šar and T. Gologranc, On a local 3-Steiner convexity, European J. Combin., 32 (2011) 1222-1235.
  • J. Cáceres and O. R. Oellermann, On 3-Steiner simplicial orderings, Discrete Math., 309 (2009) 5828-5833.
  • J. Cáceres, O. R. Oellermann and M. L. Puertas, Minimal trees and monophonic convexity, Discuss. Math. Graph Theory, 32 (2012) 695-704.
  • M. Changat, H. M. Mulder and G. Sierksma, Convexities related to path properties on graphs, Discrete Math., 290 (2005), 117-131.
  • F. F. Dragan, F. Nicolai and A. Brandstädt, Convexity and HHD-free graphs, SIAM J. Discrete Math., 12 (1999) 119-135.
  • M. Henning, M. H. Nielsen and O. R. Oellermann, Local Steiner convexity, European J. Combin., 30 (2009) 1186-1193.
  • M. Farber and R. E. Jamison, Convexity in graphs and hypergraphs, SIAM J. Algebr. Discrete Methods, 7 (1986) 433-444.
  • M. Farber and R. E. Jamison, On local convexity in graphs, Discrete Math., 66 (1987) 231-247.
  • E. Kubicka, G. Kubicki and O. R. Oellermann, Steiner intervals in graphs, Discrete Math., 81 (1998) 181-190.
  • M. H. Nielsen and O. R. Oellermann, Local 3-monophonic convexity, J. Combin. Math. and Combin. Comput., 80 (2012) 11-24.
  • M. H. Nielsen and O. R. Oellermann, Steiner trees and convex geometries, SIAM J. Discrete Math., 23 (2009) 680-693.
  • O. R. Oellermann and M. L. Puertas, Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs, Discrete Math., 307 (2007) 88-96.
  • M. J. L. van de Vel, Theory of Convex Structures, Amsterdam, North-Holland, 1993.