Some Morphological Operators in Graph Spaces

We study some basic morphological operators acting on the lattice of all subgraphs of a (non-weighted) graph $\mathbb{G}$. To this end, we consider two dual adjunctions between the edge set and the vertex set of $\mathbb{G}$. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of $\mathbb{G}$ and (ii) to extend it to subgraphs of $\mathbb{G}$. Afterward, we propose several new erosions, dilations, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of $\mathbb{G}$ and (ii) on the subgraphs of $\mathbb{G}$.