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In this paper, we consider the notion of \emph{special treewidth}, recentlyintroduced by Courcelle\cite{Courcelle2012}. In a special tree decomposition,for each vertex $v$ in a given graph, the bags containing $v$ form a rootedpath. We show that the class of graphs of special treewidth at most two isclosed under taking minors, and give the complete list of the six minorobstructions. As an intermediate result, we prove that every connected graph ofspecial treewidth at most two can be constructed by arranging blocks of specialtreewidth at most two in a specific tree-like fashion.  Inspired from the notion of special treewidth, we introduce three naturalvariants of treewidth, namely \emph{spaghetti treewidth}, \emph{stronglychordal treewidth} and \emph{directed spaghetti treewidth}. All theseparameters lie between pathwidth and treewidth, and we provide commonstructural properties on these parameters. For each parameter, we prove thatthe class of graphs having the parameter at most two is minor closed, and wecharacterize those classes in terms of a \emph{tree of cycles} with additionalconditions. Finally, we show that for each $k\geq 3$, the class of graphs withspecial treewidth, spaghetti treewidth, directed spaghetti treewidth, orstrongly chordal treewidth, respectively at most $k$, is not closed undertaking minors.

Characterizing width two for variants of treewidth