  Since the introduction of binomial edge ideals $J_{G}$ by Herzog et al. and
independently Ohtani, there has been significant interest in relating algebraic
invariants of the binomial edge ideal with combinatorial invariants of the
underlying graph $G$. Here, we take up a question considered by Herzog and
Rinaldo regarding Castelnuovo--Mumford regularity of block graphs. To this end,
we introduce a new invariant $\nu(G)$ associated to any simple graph $G$,
defined as the size of a particularly nice induced maximal matching of
$\text{in}_{\text{lex}}(J_{G})$. We investigate the question of expressing
$\nu(G)$ in terms of the combinatorics of $G$. We prove that $\nu(G)$ is the
maximal total length of a certain collection of induced paths within $G$. We
prove that for any graph $G$, $\nu(G) \leq \text{reg}(J_{G})-1$, and that the
length of a longest induced path of $G$ is less than or equal to $\nu(G)$; this
refines an inequality of Matsuda and Murai. We then investigate the question:
when is $\nu(G) = \text{reg}(J_{G})-1$? We prove that equality holds when $G$
is closed; this gives a new characterization of a result of Ene and Zarojanu,
and when $G$ is bipartite and $J_{G}$ is Cohen-Macaulay; this gives a new
characterization of a result of Jayanathan and Kumar. For a block graph $G$ we
prove that $\nu(G)$ admits a combinatorial characterization independent of any
auxiliary choices, and we prove that $\nu(G) = \text{reg}(J_{G})-1$. This gives
$\text{reg}(J_{G})$ a combinatorial interpretation for block graphs, and thus
answers the question of Herzog and Rinaldo.
