Monotone maps, the likeness relation and \(G\)-structures (Q952607)

For any graph \(G\), the authors introduce the notion of \(G\)-structure on a metric continuum \(X\) as a generalization of a linear or circular chain cover of \(X\) by its subcontinua. The main results in the paper are: (a) A locally connected continuum has a \(G\)-structure if and only if \(G\) is \(X\)-like; (b) Any nondegenerate locally connected continuum has an arc-structure or a circle-like structure; (c) A locally connected connected continuum \(X\) can be monotonely mapped onto a graph if and only if \(X\) contains uncountably many disjoint local separating continua. The authors also offer some invariants of the likeness relation and include the following problem: Suppose that \(Y\) is a nondegenerate continuum such that for each continuous image \(X\) of \(Y\) we have that \(Y\) is \(X\)-like. Is \(Y\) a pseudo-arc?