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Toll number of the strong product of graphs. (English) Zbl 1403.05122
Summary: A tolled walk $$T$$ between two non-adjacent vertices $$u$$ and $$v$$ in a graph $$G$$ is a walk, in which $$u$$ is adjacent only to the second vertex of $$T$$ and $$v$$ is adjacent only to the second-to-last vertex of $$T$$. A toll interval between $$u, v \in V(G)$$ is a set $$T_G(u, v) = \{x \in V(G) \mid x \text{ lies on a tolled walk between } u \text{ and } v \}$$. A set $$S \subseteq V(G)$$ is toll convex, if $$T_G(u, v) \subseteq S$$ for all $$u, v \in S$$. A toll closure of a set $$S \subseteq V(G)$$ is the union of toll intervals between all pairs of vertices from $$S$$. The size of a smallest set $$S$$ whose toll closure is the whole vertex set is called the toll number of $$G$$, $$\operatorname{tn}(G)$$. This paper investigates the toll number of the strong product of graphs. First, a description of toll intervals between two vertices in the strong product graphs is given. Using this result we characterize graphs with $$\operatorname{tn}(G \boxtimes H) = 2$$ and graphs with $$\operatorname{tn}(G \boxtimes H) = 3$$, which are the only two possibilities. As an addition, for the t-hull number of $$G \boxtimes H$$ we show that $$\operatorname{th}(G \boxtimes H) = 2$$ for any non-complete graphs $$G$$ and $$H$$. As extreme vertices play an important role in different convexity types, we show that no vertex of the strong product graph of two non-complete graphs is an extreme vertex with respect to the toll convexity.
MSC:
 05C76 Graph operations (line graphs, products, etc.)
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