Typically, so-called "dominant" participle constructions (aka Ab urbe condita constructions; AUC for short) are defined by saying that the predicative participle is compulsory, whereby it cannot be eliminated without a significant change of meaning (e.g., (1a)). In some cases its elimination can be said to give semantic anomaly (e.g., (1b)).
(1) a. Ab urbe condita vs. Ab urbe (condita) ‘Since the foundation of the city’ // 'From the city (which was founded)'
b. Virtus constat ex hominibus tuendis (Cic. Off. 1.157) vs. Virtus constat ex hominibus (tuendis) ‘Virtue centres in protecting people’ // #'Virtue centres in people (who must be protected').
However, there are some cases where the participle/secondary predicate in AUC constructions becomes optional, although perhaps in different degrees (e.g., cf. (2a) and (2b)). Is there any relevant generalization to be made here or each case has its own story?
(2) a. Ante Christum natum
b. Post Ciceronem consulem
Indeed, it is not always easy to work out if the participle is dominant/compulsory or optional: e.g., vid. (3a) (cf. (3b), where servandum is a final/purpose optional adjunct). For some related discussion, see How many types of so-called “predicative Gerundives” can be differentiated in Latin?
(3) a. Populus Romanus consuli potius Crasso quam privato Africano bellum gerendum dedit (Cic. Phil. 11.18) ‘The Roman people preferred the consul C. to the private person A. as their leader in the war’ (NB: Pinkster's (1990) translation).
b. Diviti homini id aurum servandum dedit (Pl. Bacch. 338) ‘He gave that gold to a rich man to keep’.
EDIT 1 (thanks to Joonas's excellent answer below)
Notice that there seems to be a grammatical distinction to be made between (3a) and (3b): (3a) can be considered an AUC construction to the extent that the predicative gerundive is part of the argumental direct object of dedit: as pointed out by Joonas, what was given was not war but the leadership of war. In contrast, in (3b) what was given was a thing (gold), whereby servandum is to be considered as a predicative adjunct outside of the argumental direct object (id aurum). Note this is not an obstacle for the grammatically optional predicative adjunct servandum to be considered as obligatory for informational reasons (as for so-called "obligatory adjuncts", feel free to google these terms: e.g., Engl. These horses saddle easily. "Easily" is a a grammatical adjunct but cannot be eliminated due to informational/pragmatic reasons: cf. ??These horses saddle).
Accordingly, to the extent that (3a) contains an AUC construction, note that this example (3a) is more similar to (3c) rather than to (3b):
(3) c.Epaminondam pecunia corrumpendum suscepit (Nep. 15,4,1) ‘He undertook the bribing of Epaminondas with money’.
It is also worth pointing out another insightful remark made by Joonas. See his very interesting connection between the so-called AUC construction and what he refers to as a cognitive principle known as metonymy. It's really fascinating for me to see that his insightful intuition is also found in the specialized literature on the topic of dominant participles & gerundives (e.g., see http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/24318 ).
END EDIT 1
One could conclude that the distinction found in Latin grammar textbooks between dominant/obligatory predicates and optional ones is not “real” but is a by-product from our attempt of imposing the grammatical schemata of our native languages (e.g., the ones of English, Catalan, etc). However, I don’t think this relativistic point is correct but rather it seems to me that there’s something grammatically real behind the (allegedly) universal syntactic distinction between being an obligatory secondary predicate and being an optional attributive modifier.
Finally, I was wondering if there are meaning differences between the following pairs. Or are they merely "stylistic"?
(4) a. Post Ciceronem consulem // Post Ciceronis consulatum
b. Ab urbe condita // Ab urbis conditione