Haven (graph theory)

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In graph theory, if $G$ is a graph, and $k \ge 0$ is an integer, a haven of order $k$ in $G$ is a function assigning to every set $X \subseteq V(G)$ with $\big|X\big| < k$ a vertex set of a component of $G \big\backslash X$ , $\beta\big(X\big)$ , such that if $X \subseteq Y \subseteq V(G)$ and $\big|Y\big| < k$ , then $\beta(Y) \subseteq \beta(X)$ .