Breadth is depth
Suppose you put bacteria in the center of a petri dish covered in agar. Over time, the bacteria will pervasively spread from the center. Though growing in infinite coordinate directions, it's simply filling the dish, working towards the edge—so equally a singular direction: outwardly.
To preface, depth & breadth -first search are fundamentally different algorithms and do not directly apply to my saying, but they serve as the foundation to my point. Depth-first search is a graph traversal algorithm where one explores a single branch as deeply as possible before backtracking to explore other paths. In contrast, with breadth-first search, one recursively explores the neighbors of what has previously been, working outward and, in the process, parsing the graph into layers by distance from the starting vertex. Each layer is a subgraph: a non-empty set of vertices and set of edges within. The additional "property" to a layer is the non-empty set of edges that connects it to the former & latter.
Tangentially, it's often impossible to use mathematical objects to perfectly describe something in reality. In practice, graphs are data structures, and their vertices have a finite set of fields to store information. So not only is there quality to how vertices are connected but also to the vertex itself. But at what granularity do we say we've stored enough, or maybe the maximal amount of, information to encapsulate something? Suppose we were to represent your knowledge base right now as a graph: is it right to let a memory be a vertex? Or through a macro lens, would a topic or time period be a better assignment? A subidea within a memory? A neuron?
Layers have properties describing how they're intraconnected & interconnected. Could we let these properties just be fields in a vertex? Conversely, is there too much detail lost by fitting the complexity of a quality to a field that the vertex might be better represented as its own graph?
Continuing on the former thought, suppose we successfully compact each layer of a graph to a vertex; we form what is known as a path graph: ._._._ ... _. When starting a searching algorithm at the vertex in the path graph that corresponds to the initial layer, we're starting at a leaf/endpoint. Now, there's only one branch to explore, and each layer is already a vertex, so breadth & depth -first search perform identical traversals.
Exploring a lot of different things, at the end of the day, is just exploring. All the paths I trek is but a single tour. I say breadth is depth so you question the vertex. What is it, and what liberty do you have over defining it?
Because ultimately, the quality of vertex is what matters. The way in which you think about doing is pointless if you don't do meaningful things.