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

.