Schaudt, Oliver and Schrader, Rainer and Weil, Vera
On the separability of graphs.
Discrete Mathematics Vol. 313 (6).
Recently, Cicalese and Milanič introduced a graph-theoretic concept called separability. A graph is said to be k-separable if any two non-adjacent vertices can be separated by the removal of at most k vertices. The separability of a graph G is the least k for which G is k-separable. In this paper, we investigate this concept under the following three aspects. First, we characterize the graphs for which in any non-complete connected induced subgraph the connectivity equals the separability, so-called separability-perfect graphs. We list the minimal forbidden induced
subgraphs of this condition and derive a complete description of the
separability-perfect graphs.We then turn our attention to graphs for which the separability is given locally by the maximum intersection of the neighborhoods of any two non-adjacent vertices. We prove that all (house,hole)-free graphs fulfill this property – a class properly including the chordal graphs and the distance-hereditary graphs. We conclude that the separability can be computed in O(m∆) time for such graphs.In the last part we introduce the concept of edge-separability, in analogy to edge-connectivity, and prove that the class of k-edge-separable graphs is closed under topological minors for any k. We explicitly give the forbidden topological minors of the k-edge-separable graphs for each 0 ≤ k ≤ 3.
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