My guess: a "deformation preserving dual graph" $\tilde C$ of $C$ over some Artinian local $A$ is a flat lifting $\tilde C/A$ which formally locally looks like ${\rm Spec}(A[x,y]/(xy))$. Some would maybe call it an equisingular deformation. This is to be contrasted with "smoothing" deformations, which locally can look like ${\rm Spec}(A[x,y]/(xy-a)$ for some $a$ in the maximal ideal of $A$. In simpler terms, this is just a deformation of the normalization $C'$ of $C$ together with a deformation of the divisor $C' \times_C C'$. If the components $C_i$ of $C$ are smooth (no self-intersections), this means deforming each $C_i$ separately and then deforming the intersection points $p\in C_i\cap C_j$ both as a point in $C_i$ and $C_j$. For example, if each $C_i$ has genus zero and hence no non-trivial deformations, a first order deformation of $C$ preserving the dual graph means choosing a tangent vector to each component at each singular point, modulo global vector fields on each $C_i$. Since $H^0(\mathbf{P}^1, T)$ is three-dimensional, this means that the dimension of such deformations equals $\sum \max(0, n_i - 3)$ where $n_i$ is the number of singular points on $C_i$ (self-intersections being counted twice), see the formula on p. 29.

In less concrete but more concise terms, such an "equisingular" curve is (probably) a flat and proper family over some base whose fibers have at worst nodal singularities and whose singular locus is also flat.

(I don't know about stable maps.)