The ticketing carrier must have agreements with every other carrier.
The ticketing carrier is the carrier on whose ticket paper the itinerary is printed. In modern times, it is the carrier whose server holds the e-ticket. The ticketing carrier is also known as the validating carrier or the plating carrier, essentially synonymously.
The ticketing carrier takes payment from the travel agent, or the purchaser directly, and is responsible for distributing those funds to the marketing carriers on the ticket as appropriate. It is also responsible for handling cancellations, changes and other sundry matters to do with the ticket.
In the case of electronic tickets, the ticketing carrier also transfers control of the appropriate flight coupon directly to the operating carrier at check in. (Each flight coupon is valid for one flight; in the old days a coupon was torn off the ticket booklet and stuck in your boarding pass at check in; the carrier then took the coupon from you as you boarded. Today this happens electronically.) The flight coupon is a financial instrument representing proof that the carrier has fulfilled its obligation to transport the passenger. In the event that that the ticketing carrier does not have such an agreement with the operating carrier, perhaps following a ticket change, the ticket will have to be printed to paper so that the flight coupon can be physically given to the operating carrier. This rarely (probably never) happens now.
For these reasons, the ticketing carrier needs to have ticketing interline agreements with each of the operating and marketing carriers on the itinerary.
(That doesn't include wetlease providers who provide aircraft and crew to a network airline (either permanently or temporarily), but it does include franchise carriers who trade under another carrier's name but are independent entities.)
Usually the ticketing carrier will be the marketing carrier who offers carriage on the longest leg, but there is no reason that has to be the case. If you buy a ticket directly from an airline, that airline will always be the ticketing carrier.
Each marketing carrier must have an interline agreement with the carrier(s) operating its flights.
A one-way interline agreement is a prerequisite of a codeshare agreement. But for completeness, each marketing carrier must interline with the true operating carrier that it markets.
Further interline agreements are helpful, but not usually necessary.
This is the situation where you have carrier XX followed by carrier ZZ, ticketed by TT who interlines with both, but XX and ZZ do not interline with each other. This is possible, but: XX cannot check you in for the ZZ sector, there will be no baggage transfer from XX and ZZ, and ZZ will have no notice if you are delayed or inconvenienced by XX. Even worse, XX has no legal authority to alter your ZZ flight coupon and so no ability to re-accommodate you on a later ZZ flight, and may refer you back to TT to make further changes to the ticket. I think most travel agents would refuse to issue a ticket containing a connection between two non-interlining carriers.
The only time I really see this situation is on a round the world trip where one tiny regional carrier is offering a short leg somewhere, and another carrier sometime later on the ticket, on the other side of the globe, does not interline with them.
Source: IATA Ticketing Handbook
On fare compatibility.
Fares often impose restrictions on who the ticketing carrier may be, for instance Delta transatlantic fares can only be issued on Delta, Air France, Virgin Atlantic, or KLM ticket paper. If you wanted to combine that with some other fare, the second fare must be okay with this restriction. If your new fare had to be issued on Alaska ticket stock, this would not be possible on the same ticket.