A note on PM-compact $K_4$-free bricks

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching covered graphs. Lovász (Combinatorica, 3 (1983), 105-117) showed that every brick is $K_4$-based or ${\overline{C}}_6$-based. A brick is $K_4$-free (respectively, ${\overline{C}}_6$-free) if it is not $K_4$-based (respectively, ${\overline{C}}_6$-based). Recently, Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 34(3) (2020), 1769-1790) characterised the PM-compact ${\overline{C}}_6$-free bricks. In this note, we show that, by using the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769-817), the only PM-compact $K_4$-free brick is ${\overline{C}}_6$, up to multiple edges.