Conditional Invariants and Interpolation Theorems for Graphs

Abstract

Let $\mathcal{F}$ be a family of objects and $\phi$ an integer-valued function defined on $\mathcal{F}$.If for any $A,B\in \mathcal{F}$ and integer $k$ between $\phi (A)$ and $\phi (B)$, there exists $C\in \mathcal{F}$ such that $\phi (C)=k$, then $\phi$ 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.