Intersection graphs of L-shapes and segments in the plane
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An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx1.jpg">, mmlsi46" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si46.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=205ef0a975f60149046cce01417dd572" title="Click to view the MathML source">mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx2.jpg">,mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx3.jpg">mathContainer hidden">mathCode"><math altimg="si46.gif" overflow="scroll"><mtext>mtext><mo>,mo><mtext>mtext>math> and mmlsi47" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si47.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=939a53f2ef37cfc2cd80360ac4d6eed7" title="Click to view the MathML source">mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx4.jpg">mathContainer hidden">mathCode"><math altimg="si47.gif" overflow="scroll"><mtext>mtext>math>. 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Motivated by a theorem of Middendorf and Pfeiffer (1992), stating that every {mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx1.jpg">,mmlsi47" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si47.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=939a53f2ef37cfc2cd80360ac4d6eed7" title="Click to view the MathML source">mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx2.jpg">mathContainer hidden">mathCode"><math altimg="si47.gif" overflow="scroll"><mtext>mtext>math>}-graph is a mmlsi55" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si55.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=201494e2751c0048bba8cfebd830e60a" title="Click to view the MathML source">SEGmathContainer hidden">mathCode"><math altimg="si55.gif" overflow="scroll"><mo>SEGmo>math>-graph, we investigate several known subclasses of mmlsi55" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si55.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=201494e2751c0048bba8cfebd830e60a" title="Click to view the MathML source">SEGmathContainer hidden">mathCode"><math altimg="si55.gif" overflow="scroll"><mo>SEGmo>math>-graphs and show that they are {mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx1.jpg">}-graphs, or mmlsi53" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si53.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=cd50822141122f79671cde8fb8c55934" title="Click to view the MathML source">BkmathContainer hidden">mathCode"><math altimg="si53.gif" overflow="scroll"><msub><mrow><mi>Bmi>mrow><mrow><mi>kmi>mrow>msub>math>-mmlsi17" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si17.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=b8135365a3e46fc4ea177bd035580f97" title="Click to view the MathML source">VPGmathContainer hidden">mathCode"><math altimg="si17.gif" overflow="scroll"><mo>VPGmo>math>-graphs for some small constant mmlsi48" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16300154&_mathId=si48.gif&_user=111111111&_pii=S0166218X16300154&_rdoc=1&_issn=0166218X&md5=bc07445056c89fd0a3e6c96e0227c59d" title="Click to view the MathML source">kmathContainer hidden">mathCode"><math altimg="si48.gif" overflow="scroll"><mi>kmi>math>. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {mg class="imgLazyJSB inlineImage" height="8" width="5" alt="Full-size image (25 K)" title="Full-size image (25 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16300154-fx1.jpg">}-graphs. 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Here a full subdivision is a graph in which each edge is subdivided at least once.

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