Leaps: an approach to the block structure of a graph

To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation ${\displaystyle {+}_G}$ as well as the set of leaps ${\displaystyle L_G}$ of the connected graph G. The underlying graph of ${\displaystyle {+}_G}$, as well as that of ${\displaystyle L_G}$, turns out to be just the block closure of G (i.e., the graph obtained by making each block of G into a complete subgraph).