How can a developer establish communication between components that are not in the same DOM...

To establish communication between components that are not in the same DOM (Document Object Model) tree, a developer can use the publish-subscribe pattern. The publish-subscribe pattern is a messaging pattern thatallows components to communicate with each other without being directly connected or aware of each other. The components can publish events to a common channel and subscribe to events from that channel. The channel acts as a mediator that delivers the events to the subscribers. The developer can use a custom library or a Salesforce platform service, such as Lightning Message Service or Platform Events, to implement the publish-subscribe pattern. Configuring the targets property is not a way to communicate between components that are not in the same DOM tree, as it only defines where a component can be used in an app. Using dispatch events is not a way either, as it only works for components that are in the same DOM tree or have a parent-child relationship. Using @api decorators is not a way either, as it only exposes public properties or methods of a component to other components that use it. Salesforce References: Lightning Web Components Developer Guide: Communicate AcrossSalesforce UI Technologies, Lightning Web Components Developer Guide: Communicate with Events, [Lightning Web Components Developer Guide: Communicate with Properties]