Reachability.2C transitive closure.2C and transitive reduction Directed acyclic graph

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the transitive reduction of dag g graph fewest edges represents same reachability relation g. subgraph of g, formed discarding edges u → v g contains longer path connecting same 2 vertices. transitive closure, transitive reduction uniquely defined dags. in contrast, directed graph not acyclic, there can more 1 minimal subgraph same reachability relation.



a hasse diagram representing partial order of set inclusion (⊆) among subsets of three-element set.


if dag g has reachability relation described partial order ≤, transitive reduction of g subgraph of g has edge u → v every pair in covering relation of ≤. transitive reductions useful in visualizing partial orders represent, because have fewer edges other graphs representing same orders , therefore lead simpler graph drawings. hasse diagram of partial order drawing of transitive reduction in orientation of each edge shown placing starting vertex of edge in lower position ending vertex.








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