# Steiner convex sets of grids

### 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.

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