# Independence-Domination duality in weighted graphs

## TL;DR

This paper explores a duality between independence and domination in weighted graphs, extending known results to a weighted setting and contributing to a broader conjecture about this duality.

## Contribution

It proves a weighted version of a known independence-domination duality result, advancing understanding of this relationship in weighted graphs.

## Key findings

- Established a weighted independence-domination duality theorem
- Extended the known unweighted results to weighted graphs
- Contributed to the broader conjecture on duality in graph theory

## Abstract

Given a partition ${\mathcal V}=(V_1, \ldots,V_m)$ of the vertex set of a graph $G$, an {\em independent transversal} (IT) is an independent set in $G$ that contains one vertex from each $V_i$. A {\em fractional IT} is a non-negative real valued function on $V(G)$ that represents each part with total weight at least $1$, and belongs as a vector to the convex hull of the incidence vectors of independent sets in the graph. It is known that if the domination number of the graph induced on the union of every $k$ parts $V_i$ is at least $k$, then there is a fractional IT. We prove a weighted version of this result. This is a special case of a general conjecture, on the weighted version of a duality phenomenon, between independence and domination in pairs of graphs.