Transitive closure and transitive reduction in bidirected graphs

TL;DR

This paper extends the concepts of transitive closure and reduction from directed graphs to bidirected graphs, introducing new notions like bipath and bicircuit, and explores their relation to matroids of signed graphs.

Contribution

It generalizes transitive closure and reduction to bidirected graphs using bipaths and bicircuits, linking these to matroid theory.

Findings

Defined bipath and bicircuit in bidirected graphs

Established relationships between transitive reduction and closure in bidirected graphs

Connected transitive concepts to matroids of signed graphs

Abstract

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.