Characterizations of indecomposable subcontinua of graph-like continua

In [Fund. Math. 247 (2019), 131–149] we introduced the notion of tracing property (called there “free tracing property”) by free $G$-chains of a $G$-like continuum $X$, and showed that if $X$ has a Cantor set with this property, then $X$ contains an indecomposable subcontinuum. Here we establish the converse, yielding the following characterization theorem.

$\mathbf{Theorem.}$ Suppose that $G$ is a graph and $H$ is a subcontinuum of a $G$-like continuum $X$. Then $H$ is indecomposable if and only if there is a Cantor set $Z$ in $H$ such that $Z$ has the tracing property by free $G$-chains and $H$ is the unique minimal continuum in $X$ containing $Z$.

Also, for tree-like continua and arc-like continua, we study the relations between “tracing property” and “composant-uniqueness”.