Abstract
The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. Broere et al. proved that if G is a graph of order n, then \(\sqrt n \leqslant \rho \left( G \right) + \rho \left( {\bar G} \right) \leqslant \left\lceil {\tfrac{{3n}} {2}} \right\rceil\). In this paper, we characterize the graphs G for which \(\rho \left( G \right) + \rho \left( {\bar G} \right) = \left\lceil {\tfrac{{3n}} {2}} \right\rceil\), improve the lower bound on \(\rho \left( G \right) + \rho \left( {\bar G} \right)\) by one when n is the square of an odd integer, and determine a best possible upper bound for \(\rho \left( G \right) + \rho \left( {\bar G} \right)\) when neither G nor \(\bar G\) has isolated vertices.
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Hattingh, J.H., Saleh, O.A., van Der Merwe, L.C. et al. Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement. Acta. Math. Sin.-English Ser. 28, 2365–2372 (2012). https://doi.org/10.1007/s10114-012-0727-6
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DOI: https://doi.org/10.1007/s10114-012-0727-6