Intersection Graphs of Maximal Sub-polygons of k-Lizards

Abstract

Abstract 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 $$k>1$$ k > 1 ; trees are 2-MSP graphs; trees are k-MSP graphs for $$k>2$$ k > 2 if and only if they are caterpillars; and n-cycles are not k-MSP graphs for $$n>3$$ n > 3 and $$k>1$$ 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 $$k>1$$ k > 1 , $$j>1$$ j > 1 , $$k \ne j$$ k ≠ j