Intersection Graphs of Maximal Sub-polygons of $k$-Lizards

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 $k>1$, $j>1$, $k \neq j$.