# Steiner convex sets of grids

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Tanja Gologranc

Faculty of Natural Sciences and Mathematics, University of Maribor

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**Minisymposium:**
GRAPH PRODUCTS

**Content:**
The Steiner tree of a (multi)set of vertices $R =\{u_1,\ldots ,u_k\} \subseteq V(G)$ is the smallest
tree in $G$ that contains all vertices of $R$. The Steiner distance $d(R)$ of a set $R$ is the number of edges in a Steiner tree $T$ for $R$. The $k$-Steiner interval of a set $R \subseteq V(G)$,
$I(R),$ consists of all vertices in $G$ that lie on some Steiner tree for $R$.
A set $S$ of vertices is $k$-Steiner convex, if the Steiner interval $I(R)$ of every (multi)set $R$ on $k$ vertices is contained in $S$. We say that a set $S$ is Steiner convex if it is $k$-Steiner convex, for every $k \geq 2.$ In this talk the characterization of Steiner convex sets of grids will be presented.