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Path extendability of $$s$$-vertex connected graphs. (English) Zbl 1240.05164
Summary: Let $$G$$ be a simple graph. $$G$$ is called $$s$$-vertex connected if every induced subgraph with $$s$$ vertices is connected but there exists an induced subgraph with $$s-1$$ vertices that is not connected, where $$s\geq 3$$. A path $$P$$ is extendable if there exists another path $$P'$$ such that $$V (P')\supset V (P)$$ and $$|V (P')|=|V (P)|+1$$. A graph $$G$$ is said to be fully path extendable if its diameter is at most two and every path $$P$$ with $$|V (P)|<|V (G)|$$ is extendable. In this paper, we prove that an $$s$$-vertex connected graph whose order is not less than $$2s-1$$ is fully path extendable.
##### MSC:
 05C38 Paths and cycles 05C40 Connectivity