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.
Citation
Tanja Gologranc. "GRAPHS WITH 4-STEINER CONVEX BALLS." Taiwanese J. Math. 19 (5) 1325 - 1340, 2015. https://doi.org/10.11650/tjm.19.2015.4403
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