  We introduce $k$-maximal sub-polygon graphs ($k$-MSP graphs), the
intersection graphs of maximal polygons contained in a polygon with sides
parallel to a regular $2k$-gon. We prove that all complete graphs are $k$-MSP
graphs for all $k>1$; trees are $2$-MSP graphs; trees are $k$-MSP graphs for
$k>2$ if and only if they're caterpillars; and $n$-cycles are not $k$-MSP
graphs for $n>3$ and $k>1$. We derive bounds for which $j$-cycles appear as
induced subgraphs of $k$-MSP graphs. As our main result, we construct examples
of graphs which are $k$-MSP graphs and not $j$-MSP graphs for all $k>1$, $j>1$,
$k \neq j$.
