Conditional Invariants and Interpolation Theorems for Graphs

Let \(\mathcal{F}\) be a family of objects and \(\varphi\) an integer-valued function defined on \(\mathcal{F}\).
If for any \(A, B \in \mathcal{F}\) and integer \(k\) between \(\varphi(A)\) and \(\varphi(B)\), there exists \(C \in \mathcal{F}\) such that \(\varphi(C) = k\), then \(\varphi\) is said to interpolate over \(\mathcal{F}\).
In this paper, we first discuss some basic ideas used in proving interpolation theorems for graphs.
By using this, we then prove that a number of conditional invariants interpolate over some families of subgraphs of a given connected graph.