K4-free planar minimal bricks with the maximum number of edges

Abstract

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick $G$ is minimal if $G-e$ is not a brick, for every edge $e$ of $G$. Lovász (1983) 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). Kothari and Murty (2016) proved that a planar brick is $K_4$-free if and only if it has precisely two odd faces and determined the list of all ${\overline{C}}_6$-free planar bricks. In this paper, we show that the $K_4$-free planar (minimal) bricks $G$ have at most $2\left|V\left(G\right)\right|-3$ edges. Furthermore, we characterize all the extremal graphs that meet this upper bound.