Geodesic
Transitive closure and transitive reduction in bidirected graphs
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 295-315.

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In a bidirected graph, an edge has a direction at each end, so bidirected graphs generalize directed graphs. We generalize the definitions of transitive closure and transitive reduction from directed graphs to bidirected graphs by introducing new notions of bipath and bicircuit that generalize directed paths and cycles. We show how transitive reduction is related to transitive closure and to the matroids of the signed graph corresponding to the bidirected graph.
DOI : 10.21136/CMJ.2019.0644-16
Classification : 05C20, 05C22, 05C38
Keywords: bidirected graph; signed graph; matroid; transitive closure; transitive reduction 
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     title = {Transitive closure and transitive reduction in bidirected graphs},
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Bessouf, Ouahiba; Khelladi, Abdelkader; Zaslavsky, Thomas. Transitive closure and transitive reduction in bidirected graphs. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 295-315. doi : 10.21136/CMJ.2019.0644-16. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0644-16/

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