Component-trees model the structure of grey-level images by considering their binary level-sets obtained from successive thresholdings. They also enable to define anti-extensive filtering procedures for such images. We generalise the notion of component-tree, and its associated filtering framework, to deal with any (grey-level or multivalued) images. This work is divided in two articles. In this first article, we define a new data structure, the component-graph, which extends the notion of component-tree to images taking their values in any (partially or totally) ordered set. The component-graphs are declined in three variants, of increasing richness and size, whose structural properties are studied. In the second article, the standard filtering framework based on component-trees is generalised to component-graphs, thus relaxing the constraints linked to total orderings on image values. Some application examples are also proposed in this second article, illustrating the potential usefulness of component-graphs in the field of multivalued image processing.