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Reachability : ウィキペディア英語版
Reachability
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. We say that a vertex s can reach a vertex t (or that t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a path) which starts with s and ends with t.
In an undirected graph, it is sufficient to identify the connected components, as any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component. The connected components of a graph can be identified in linear time. The remainder of this article focuses on reachability in a ''directed'' graph setting.
== Definition ==
For a directed graph G = (V, E), with vertex set V and edge set E, the reachability relation of G is the transitive closure of E, which is to say the set of all ordered pairs (s,t) of vertices in V for which there exists a sequence of vertices v_0 = s, v_1, v_2, ..., v_k = t such that the edge (v_,v_i) is in E for all 1 \leq i \leq k.〔.〕
If G is acyclic, then its reachability relation is a partial order; any partial order may be defined in this way, for instance as the reachability relation of its transitive reduction.〔.〕
A noteworthy consequence of this is that since partial orders are anti-symmetric, if s can reach t, then we know that t ''cannot'' reach s. Intuitively, if we could travel from s to t and back to s, then G would contain a cycle, contradicting that it is acyclic.
If G is directed but ''not'' acyclic (i.e. it contains at least one cycle), then its reachability relation will correspond to a preorder instead of a partial order.
〔.〕

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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