Transitive reductionIn the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges as possible, such that for all pairs of vertices v, w a (directed) path from v to w in D exists if and only if such a path exists in the reduction. Transitive reductions were introduced by , who provided tight bounds on the computational complexity of constructing them. More technically, the reduction is a directed graph that has the same reachability relation as D.
PolytreeIn mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest.
ReachabilityIn graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex can reach a vertex (and is reachable from ) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with and ends with . In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric ( reaches iff reaches ).
Meyniel graphIn graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them. The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they are perfect graphs in 1976, long before the proof of the strong perfect graph theorem completely characterized the perfect graphs.
