## Taiwanese Journal of Mathematics

### GRAPHS WITH 4-STEINER CONVEX BALLS

Tanja Gologranc

#### Abstract

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

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

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133711

Digital Object Identifier
doi:10.11650/tjm.19.2015.4403

Mathematical Reviews number (MathSciNet)
MR3412008

Zentralblatt MATH identifier
1357.05031

#### Citation

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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