On invariants of hereditary graph properties

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Abstract

The product P {ring operator} Q of graph properties P, Q is the class of all graphs having a vertex-partition into two parts inducing subgraphs with properties P and Q, respectively. For a graph invariant φ{symbol} and a graph property P we define φ{symbol} (P) as the minimum of φ{symbol} (F) taken over all minimal forbidden subgraphs F of P. An invariant of graph properties φ{symbol} is said to be additive with respect to reducible hereditary properties if φ{symbol} (P {ring operator} Q) = φ{symbol} (P) + φ{symbol} (Q) for every pair of hereditary properties P, Q. In this paper, we provide necessary and sufficient conditions for invariants to be additive with respect to reducible hereditary graph properties. We prove that the subchromatic number, the degeneracy number and tree-width and some other invariants of hereditary graph properties satisfy those conditions. © 2006 Elsevier B.V. All rights reserved.

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APA

Mihók, P., & Semanišin, G. (2007). On invariants of hereditary graph properties. Discrete Mathematics, 307(7–8), 958–963. https://doi.org/10.1016/j.disc.2005.11.048

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