Vertebrate interval graphs
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A vertebrate interval graph is an interval graph in which the maximum size of a set of independent vertices equals the number of maximal cliques. For any fixed v ≥ 1, there is a polynomial-time algorithm for deciding whether a vertebrate interval graph admits a vertex partition into two induced subgraphs with claw number at most v. In particular, when v = 2, whether a vertebrate interval graph can be partitioned into two proper interval graphs can be decided in polynomial time.
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