Could natural language parsing be a task on graphs? Maybe, at least partly.
Now, during parsing, every word contributes to some constructions. Some of these contributions are mutually exclusive. For example, in Russian a noun typically can't be nominative and accusative at the same time. So this noun's contributions for nom and acc constructions are incompatible.
Different words also may have contradicting contributions. Two nouns in the same clause can't both be accusative even if their forms are compatible with such an alternative (i.e. they both contribute to acc construction).
So here's the idea. Consider a graph whose vertices are formed by each construction contribution for each word. And where those contributions contradict each other in any way, there's an edge. The task is then to find a maximal subset of vertices which are not connected to each other.
There are, of course, other constraints. The resulting graph should make sense from the semantic viewpoint. The subset should be constructed incrementally and conservatively: if the parser can proceed without reanalysis of the already built structures, it should do so. Finally, this graph task has a very limited scope, e.g. it doesn't apply in different clauses.
That's just an idea. Currently my parser doesn't track individual contributions per construction, it just unifies them eagerly. But the test data suggests that might change.