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Covering the cliques of a graph with vertices. (English) Zbl 0766.05063
Here all graphs have order \(n\) and isolated vertics are not counted as cliques. The central problem studied is that of estimating the cardinality \(\tau_c(G)\) of the smallest set that shares a vertex with each clique of \(G\). Among other results it is shown that \(\tau_c(G)\leq n-\sqrt{2n}+{3\over 2}\) and a linear time (in the number of edges) algorithm for achieving this bound is proposed. Four associated problems are presented. For example, it is asked if \(\tau_c(G)\leq n-r(n)\) for all graphs \(G\) where \(r(n)\) is the largest integer such that every triangle-free graph contains an independent set of \(r(n)\) vertices. Also, how large triangle-free induced subgraphs does a \(K_4\)-free graph \(G\) contain.

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C35 Extremal problems in graph theory
05C85 Graph algorithms (graph-theoretic aspects)
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