Induced Minors and Region Intersection Graphs

We show that for any positive integers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> , there is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> -induced-minor-free graph of girth at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> that is not a region intersection graph over the class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> -minor-free graphs. This answers in a strong form the recently raised question of whether for every graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> there is a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> -induced-minor-free graphs are region intersection graphs over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> -minor-free graphs.