Characterization of bricks whose b-invariant edges are all solitary

Characterize bricks, other than K4, the triangular prism \(\overline{C_6}\), and the Petersen graph, in which every b-invariant edge is solitary.

Background

A brick is a 3-connected, bicritical matching-covered graph, and a removable edge is b-invariant when deleting it leaves a matching-covered graph with the same number of bricks in its tight-cut decomposition. A solitary edge is an edge contained in exactly one perfect matching.

The paper identifies a characterization problem originally proposed by Lucchesi and Murty for bricks whose b-invariant edges are all solitary, excluding $K_4$, the triangular prism $\overline{C_6}$, and the Petersen graph. The present paper studies the stronger condition that every removable edge, rather than only every b-invariant edge, is solitary; it does not present the cited characterization problem as its own resolved result.