Suppose we have a graph $G=(V,E)$ and root $r\in V$. This graph has a special property. To demonstrate the property, we label each $v_i\in V$ with the length of the shortest $r$-$v_i$ path. The special property that $G$ has is that neighbors of each vertex $v_i$ with label $l_i$ has either label $l_i+1$ or $l_i-1$. Is there a name for rooted graphs with this property?
Note that such graphs can be layered so that edges only occur as two vertices in adjacent layers.
Let's take it as given that $G$ must be connected, because otherwise we could not label every vertex as described. I claim that a graph has the mentioned property if and only if it is bipartite.
Assume first that $G$ is not bipartite. Then it must contain an odd cycle. Consider such a cycle. Obviously, the labels on adjacent vertices in this cycle cannot alternate parity, so there are two adjacent vertices on the cycle whose labels have the same parity. Therefore, $G$ cannot have the property.
Now assume that $G$ is bipartite. Obviously, the vertices are partitioned such that all the labels in the same part as $r$ are even and the other part has all of the odd labels. Consider adjacent vertices $v_i,v_j$. These vertices cannot have the same label: being adjacent they must be in different parts of the partition and thus their labels have different parities. Without loss of generality, assume $v_i<v_j$. Then there is a path of length $l_i$ from $r$ to $v_i$. Simply appending the edge $\{v_i,v_j\}$ to the end of that path gives a path of length $l_i+1$ from $r$ to $v_i$. Since we know $l_j>l_i$, this path assures us that $l_j=l_i+1$. Since $v_i,v_j$ were arbitrary adjacent vertices, all adjacent vertices in $G$ must have labels that differ by only one.