A note on PM-compact K4-free bricks - Jinqiu Zhou; Qunfang Li (AIMS Mathematics, 2022)
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.