First of all, the material parts of you cannot be utterly destroyed ie., made non-existent. The physical universe is governed by the laws of conservation of matter and energy. What does this mean? It means that the total amount of matter/energy in the universe cannot be altered, just rearranged in different forms.
So if you were destroyed all that would happen to the universe is that it would be in a different arrangement of matter and energy. However, this is what happens for every moment of the universe. It is constantly in flux. No moment of the universe is the same as any other moment. The universe is constantly rearranging itself from moment to moment.
The truth of the matter is that you are a part of the universe in the same way that your nose is a part of your body. Consider: if your nose was destroyed would your body also be destroyed? If your body was destroyed would your nose be as well? No and yes, right? However, can you alter one without altering the other? No. They are interdependent in this way. The same is true of you and the universe. And richly so... the interdependencies go far beyond this trivial level. See if you can't figure out some other interdependencies :)
In general, we sentient beings are absurdly overly confident in our own independence. Truthfully, we are so utterly dependent that any notion of independence is really foolish and comical. :)
Consider the ways in which you are dependent upon others... You are dependent on your mother, father, and their mothers, fathers, etc, going back (at least) millions of generations for your very life. Each one of those people was dependent upon myriad acts and people to sustain their lives long enough to produce offspring that eventually produced you. You depend upon countless people to produce the supply chains of food for your daily nourishment. You depend on countless construction workers and supply chains for your daily shelter. The water you drink is made possible by countless engineers and people responsible for maintaining the aquaducts and reservoirs used to provide fresh water. When you go to the grocer you depend upon all the people who worked to stock those shelves and produce the food and the packaging and the huge supply chains that went into it. Your clothing is produced by the people working the fields for cotton and other fibers. The machines used to harvest that fiber was itself designed and manufactured - by engineers and inventors - and required ore and other raw materials that itself took huge supply chains to produce. All the people above were similarly dependent on their parents for their lives. And all were dependent upon education in the various trades on their teachers. The teachers were also dependent through and through. The dependency graph is so thorough with so many nodes utterly replete with dependency that it boggles the mind!
That bug in china and you are not one. You are separate beings. However, both of you are utterly suffused with dependency and there are undoubtedly countless nodes in common between the dependency graph of that bug and your own dependency graph. It takes just a little imagination to come up with numbers of ways in which your story and that bug's are interrelated and mutually dependent.
Maybe, "you cannot alter one without altering the other" is not the definition of interdependent, but it sure seems a necessary condition! Any two things where you can truthfully say, "you cannot alter one without altering the other" you can be darn sure are interdependent :)
Another way of thinking about your question is trying to understand the relationship between a whole and the parts that make it up. Consider a car. Is a car dependent upon the things that make it up? Does the car depend upon the tires? Does it depend upon the brakes? Does it depend upon the windshield? The engine?
Say you start taking parts away from a car... at what point or at what part that you take away does it make the car no longer a car? This is a great question to consider and has been contemplated very deeply by Buddhist masters for thousands of years. Of course, earlier generations of Buddhists considered chariots not cars. The greeks considered ships. What's your answer? :)
Hope this helps!