"Complexity of $\eta$-od-like continua."
Maldonado Garcia, Hugo AdrianW. Lewis asked in Indecomposable Continua. Open problems in topology II, whether there exists, for every $\eta\geq 2$, an atriodic simple $(\eta+1)$-od-like continuum which is not simple $\eta$-od-like and, if such continuum exists, whether it has a variety of properties such as being planar or being an arc-continuum, among others. Some partial results have been obtained by W.T. Ingram, P. Minc, C.T. Kennaugh and L. Hoehn. In each case, the most substantial challenge is in proving that a continuum is not $T$-like, for a given tree $T$. We present the notion of a combinatorial $\eta$-od cover of a graph, a tool which may enable one to prove that certain examples of continua are not $\eta$-od-like. Also, we suggest the construction of an atriodic simple $(\eta+1)$-od-like continuum which is not simple $\eta$-od-like and has properties such as being planar, being an arc-continuum and span zero (This is a work in progress).