On the Anti-Kekule and Anti-Forcing Number of Cata-condensed Phenylenes

The anti-forcing number is defined as the smallest number of edges that have to be removed in order that any graph remains with a single Kekule structure. Similarly, the anti-Kekule number is defined as the smallest number of edges that have to be removed in order that any graph remains connected but without any Kekule structure. It is shown that the anti-Kekule number of cata-condensed phenylenes is 3 and the anti-forcing number of a cata-condensed [h]-phenylene is h, where h is the number of the hexagons in the cata-condensed phenylene. Moreover, it is proved that for a Kekule structure M of a cata-condensed [11]-phenylene, the forcing number phi(M) is bounded by inverted right perpendicularh/2inverted left perpendicular <= phi(M) <= h.