A graph G has a perfect matching if and only if 0 is not a root of its matching polynomial μ(G, x). Thus, Tutte's famous theorem asserts that 0 is not a root of μ(G, x) if and only if codd(G - S) ≤ |S| for all S ⊆ V(G), where codd(G) denotes the number of odd components of G. Tutte's theorem can be proved using a characterization of the structure of maximal non-matchable graphs, that is, the edge-maximal graphs among those having no perfect matching. In this paper, we prove a generalized version of Tutte's theorem in terms of avoiding any given real number Θ as a root of μ(G, x). We also extend maximal nonmatchable graphs to maximal Θ-non-matchable graphs and determine the structure of such graphs. © 2013 Elsevier B.V. All rights reserved.
Ku, C. Y., & Wong, K. B. (2013). Generalizing Tutte’s theorem and maximal non-matchable graphs. Discrete Mathematics, 313(20), 2162–2167. https://doi.org/10.1016/j.disc.2013.05.015