Abstract
Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ F (G) bounded by a constant fraction of |G|?
In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K k or \(\overline{K_k}\).
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Shen, J. Subgraph Transversal of Graphs. Graphs and Combinatorics 25, 601–609 (2009). https://doi.org/10.1007/s00373-009-0850-z
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DOI: https://doi.org/10.1007/s00373-009-0850-z