The d-blocker number and d-transversal number of hexagonal systems

Abstract

The carbon skeleton of a benzenoid hydrocarbon is often represented by a hexagonal system $H$ with a perfect matching (or Kekulé structure). Vukičević and Trinajstić defined the anti-Kekulé number $ak\left(H\right)$ as the smallest number of edges of $H$ whose removal results in a connected graph without Kekulé structures. In general, this article investigates related parameters, the $d$-blocker number ${\beta }_d\left(H\right)$ and the $d$-transversal number ${\tau }_d\left(H\right)$, introduced by Zenklusen et al. These quantify the minimum sizes of an edge subset whose removal reduces the matching number by $d$, and an edge subset that contains at least $d$ edges of every maximum matching, respectively. For parallelogram, regular hexagon-shaped, and cata-condensed hexagonal systems $H$, we derive explicit formulas for the $d$-blocker number ${\beta }_d\left(H\right)$ for all $1\le d\le \left|V\left(H\right)\right|/2$, and explicit expressions or bounds for the $d$-transversal number ${\tau }_d\left(H\right)$. These results extend the robustness analysis of Kekulé structures and generalize the matching preclusion number (corresponding to $d=1$) in benzenoid systems.