Generalized $D$-graphs for Nonzero Roots of the Matching Polynomial

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343--358) introduced a graph operator $D(G)$, called the $D$-graph of $G$, which has been useful in investigating the structural aspects of maximal Tutte sets in $G$ with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph $D(G)$ and maximal extreme sets in $G$. This was later extended to graphs without perfect matchings by Busch et al. (Discrete Appl. Math. 155 (2007), 2487--2495). Let $\theta$ be a real number and $\mu(G,x)$ be the matching polynomial of a graph $G$. Let $\textnormal{mult} (\theta, G)$ be the multiplicity of $\theta$ as a root of $\mu(G,x)$. We observe that the notion of $D$-graph is implicitly related to $\theta=0$. In this paper, we give a natural generalization of the $D$-graph of $G$ for any real number $\theta$, and denote this new operator by $D_{\theta}(G)$, so that $D_{\theta}(G)$ coincides with $D(G)$ when $\theta=0$. We prove a characterization of maximal $\theta$-Tutte sets which are $\theta$-analogue of maximal Tutte sets in $G$. In particular, we show that for any $X \subseteq V(G)$, $|X|>1$, and any real number $\theta$, $\m(\theta, G \setminus X)=\m(\theta, G)+|X|$ if and only if $\m(\theta, G \setminus uv)=\m(\theta, G)+2$ for any $u, v \in X$, $u \not = v$, thus extending the preceding work of Bauer et al. and Busch et al. which established the result for the case $\theta=0$.