Segmenting histology images into diagnostically relevant regions is imperative to support timely and reliable decisions by pathologists. To this end, computer-aided techniques have been proposed to delineate relevant regions in scanned histology slides. However, the techniques necessitate task-specific large datasets of annotated pixels, which is tedious, time-consuming, expensive, and infeasible to acquire for many histology tasks. Thus, weakly-supervised semantic segmentation techniques are proposed to leverage weak supervision which is cheaper and quicker to acquire. In this paper, we propose SEGGINI, a weakly-supervised segmentation method using graphs, that can utilize weak multiplex annotations, i.e., inexact and incomplete annotations, to segment arbitrary and large images, scaling from tissue microarray (TMA) to whole slide image (WSI). Formally, SEGGINI constructs a tissue-graph representation for an input image, where the graph nodes depict tissue regions. Then, it performs weakly-supervised segmentation via node classification by using inexact image-level labels, incomplete scribbles, or both. We evaluated SEGGINI on two public prostate cancer datasets containing TMAs and WSIs. Our method achieved state-of-the-art segmentation performance on both datasets for various annotation settings while being comparable to a pathologist baseline. Code and models are available at: https://github.com/histocartography/seg-gini.