On $\gamma$-Contraction and $\beta$-Contraction: A Unified Framework for Colour-Preserving Graph Reduction

Abstract

Graphs are a fundamental abstraction in computer science and discrete mathematics, where information is encoded in their combinatorial structure. Graph-reduction techniques aim at simplifying graphs while preserving selected structural properties, typically by grouping vertices and replacing each group with a representative, yielding a contracted graph. A common instance of this paradigm arises when vertices carry categorical information, formalised as a colouring of the vertex set. In this setting, natural contraction units correspond to connected components of vertices sharing the same colour. In this work, we provide a rigorous mathematical formalisation of $\gamma$-contraction, a colour-based graph contraction operation. We interpret $\gamma$-contraction as a quotient-like construction that preserves categorical connectivity, and clarify its relationship with classical notions of graph contraction and quotient graphs. To support a constructive and algorithmic treatment, we introduce a locally-defined iterative variant, termed $\beta$-contraction, which captures the core mechanism underlying $\gamma$-contraction. Building on this framework, we analyse the contraction process from a theoretical perspective and establish formal guarantees of correctness and convergence. In particular, we prove that $\beta$-contraction converges in a logarithmic number of iterations to $\gamma$-contraction, and that this bound is asymptotically tight, with base equal to the golden ratio.