The expansion of polynomial invariants for $2$-decompositions of generalized graphs

TL;DR

This paper extends the concept of 2-decomposition from ribbon graphs to more complex graph structures, providing new formulas for polynomial invariants that generalize previous results in graph theory.

Contribution

It generalizes 2-decomposition to half-edged ribbon graphs and rank D-weakly colored graphs, leading to new expansion formulas for associated polynomial invariants.

Findings

Extended 2-decomposition to new graph classes

Derived new expansion formulas for Bollobás-Riordan polynomial

Provided polynomial invariants for weakly colored stranded graphs

Abstract

The $2$-decomposition for ribbon graphs was introduced in [Annals of Combinatorics 15 (2011), pp 675-706]. We extend this result to half-edged ribbon graphs and to rank $D$-weakly colored graphs [SIGMA 12 (2016), 030], generalizing therefore the $2$-sums and tensor products of these graphs. Using this extension for the $2$-decompositions, we provide new expansion formulas for the Bollob\'as Riordan polynomial for half-edged ribbon graphs and also for the polynomial invariant for weakly colored stranded graphs.