The decycling number of outerplanar graphs

For a graph G, let tau(G) be the decycling number of G and c(G) be the number of vertex-disjoint cycles of G. It has been proved that c(G)a parts per thousand currency sign tau(G)a parts per thousand currency sign2c(G) for an outerplanar graph G. An outerplanar graph G is called lower-extremal if tau(G)=c(G) and upper-extremal if tau(G)=2c(G). In this paper, we provide a necessary and sufficient condition for an outerplanar graph being upper-extremal. On the other hand, we find a class of outerplanar graphs none of which is lower-extremal and show that if G has no subdivision of S for all , then G is lower-extremal.