Monotone maps, the likeness relation and G-structures

Locally connected continua which admit monotone maps onto graphs are characterized. A notion of a G-structure is introduced for any graph G as a generalization of a linear or circular chain (of arbitrary finite length) cover of a continuum by its subcontinua. It is proved that a locally connected continuum X has a G-structure iff G is X-like. We show that any nondegenerate locally connected continuum has an arc-structure or a circle-structure. We find some invariants of the likeness relation.