Abstract
For a graph G, let τ(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)≤τ(G)≤2c(G) for an outerplanar graph G. An outerplanar graph G is called lower-extremal if τ(G)=c(G) and upper-extremal if τ(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 \(\mathcal{S}\) of outerplanar graphs none of which is lower-extremal and show that if G has no subdivision of S for all \(S\in \mathcal{S}\), then G is lower-extremal.
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This research is partially supported by NSC 99-2811-M-009-056 and NSC 100-2115-M-390-004-MY2.
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Chang, H., Fu, HL. & Lien, MY. The decycling number of outerplanar graphs. J Comb Optim 25, 536–542 (2013). https://doi.org/10.1007/s10878-012-9455-1
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DOI: https://doi.org/10.1007/s10878-012-9455-1