book theory book book collection half planes having line boundary usually required lie boundary line called spine edges required stay single half plane book thickness smallest possible number half planes book book thickness called fixed book define including book crossing number book thickness formula formula gives exact book thickness complete book thickness book thickness generally book thickness minor closed families particular bounded bounded genus bounded book thickness hard determine exact book thickness given knowing fixed ordering spine book original motivations studying book involved applications design book represent components circuit wires represent connections book applications drawing standard styles arc circular constructed book transportation planning different sources destinations foot vehicle traffic meet interact traffic light modeled edges connecting different source destination pairs book design schedule lets traffic intersection signal phases possible problems involving folding structure single page book represent classical forms acid secondary structure page book represent applications book include abstract knot theory open problems concerning book thickness unknown book thickness arbitrary bounded function book thickness subdivisions testing existence page book given fixed ordering spine unknown complexity known time known hard book thickness unknown exists book thickness exactly history notion book space defined 1960s work considered books studied definition space represented distinct points edges represented curves way edges intersect meet common early 1970s developed restricted type came subsequent research placed spine book edge lie single page important later development book include proof late 1980s book thickness discovery late 1990s close connections book definitions book particular kind space called fan half planes consists single line called spine book collection half planes called pages leaves book having spine boundary books finite number pages embedded dimensional space instance choosing axis coordinate system choosing pages half planes angle respect plane multiple book drawing finite book drawing drawn point spine edge drawn curve lies single page page book crossing number minimum number crossings page book drawing book book drawing forms book drawing edge crossings finite book book large number pages instance possible edge separate page book thickness stack number minimum number pages required book measures quality book number pages maximum number edges crossed ray perpendicular spine single page book edge drawn curve maximum size edges single page intervals defined spine pairs edges intersect crucial definitions edges allowed stay single page book observed edges instead pass page spine book embedded page book page book spine crossings allowed embedded number spine crossings edge need spine crossings specific shown figure book thickness complete non book thickness greater book pages exists generally book thickness complete exactly formula result gives upper bound maximum possible book thickness page crossing number complete formula matching crossing number hill correct drawing number crossings page drawing book thickness complete min construct drawing book thickness smaller place edges incident page bound tight instance book thickness sides book thickness exactly complete formed independent sets independent set edge different independent sets book thickness formula odd upper bound improved formula book thickness shuffle exchange cube connected cycles large exactly properties book thickness given belong outer face placing order spine appear outer face provides page book given point necessarily appear ordering outer face copies included book conversely page book automatically embedded single page half plane attached spine extend page complete plane outer face includes entire added half plane lie outer face page book special case union pages book space equivalent plane book thickness automatically precisely book thickness cycle given page augmented adding page extra edges consecutive spine adjacent spine provides example book thickness possible add edges preserving cycle characterization cycles page book known maximum degree book thickness trees book thickness generally book thickness claimed exist book thickness exactly detailed proof claim announced subsequent journal paper published reason list problem determining maximum book thickness behavior subdivisions edge edge paths adding new edge increase book thickness instance diamond book thickness subdivision book thickness subdivision process significantly reduce book thickness subdivided instance book thickness complete proportional number edges edge path produces subdivision book thickness smaller formula despite existence examples subdivision book thickness smaller original specifically exists function formed replacing edge edge path book thickness book thickness remains relation book thickness related thickness number needed cover edges given thickness drawn plane edges colored colors way edges color cross book thickness drawn half plane boundary half plane edges colored colors crossing edges color book thickness colors edges correspond pages book thickness book thickness different exist subdivisions complete book thickness despite having thickness book thickness bound tight edges book thickness formula genus book thickness formula generally minor closed family bounded book thickness hand closed minors bounded book thickness including book thickness shallow minor bounded book thickness sparse ratio edges bounded constant depends depth minor book thickness terminology bounded book thickness bounded expansion bounded degree stronger requirement having bounded expansion book thickness book thickness obey removal splits pieces formula refers number exist book thickness size edges single page book behave ways like stack data structure considering arbitrary sequence push pop operations stack forming stack operations correspond placed sequence order spine book draws edge pop operation pops object stack previous push operation pushed resulting automatically page reason page number called stack number way consider arbitrary sequence operations queue data structure form operations placed order spine single page edge operation corresponding edges cross cover intervals spine analogy researchers defined queue book lies spine edge lies single page edges page cross cover intervals spine minimum number pages needed queue called queue number complexity finding book thickness hard follows fact finding cycles complete book thickness cycle exists complete test book thickness given ordering spine fixed page exists found linear time instance testing formed given cycle connecting spine ordering claimed finding page fixed spine ordering performed time result details require pages problem finding minimum possible number pages remains hard hard problem coloring circle intersection chords circle given fixed spine ordering drawing order circle drawing edges line segments produces collection chords representing form circle chords crossing pairs chords edges coloring circle represents partition edges drawn crossing single page optimal coloring equivalent optimal book circle coloring colors hard circle formed way book problem follows optimal book hard fixed ordering spine page book drawing hard minimize number crossings number spine ordering unknown partition edges pages given possible find page exists linear time based trees complete find page spine ordering edge partition known finding book crossing number hard special case testing page crossing number zero consequence bounded expansion problem finding pattern bounded size exists larger solved linear time larger bounded book thickness true detecting pattern induced larger larger reason problem testing bounded book thickness given formula order logic fixed describe system finding optimal book transforming problem instance problem applying sat resulting problem state system capable finding optimal approximately minutes successfully applied proposed requiring pages turned require pages applications fault tolerant main motivations studying book cited involves application design organization fault tolerant system developed authors system arranged logical sequence corresponding spine book sequence necessarily placed line physical layout system communication links connecting grouped correspond pages book act like connecting start new communications link pushes previous links upward bundle connecting end communication link connects bundle pops ones stack behavior single bundle handle set communications links form edges single page book organizing links way wide variety different network implemented regardless faulty long non faulty remain implement network network implemented system exactly ones book thickness equal number available book model placement wires connecting components layers circuit stack application cited concerns influential result showed system processes data stream pushing incoming elements stack appropriately chosen times stack output stream sort data initial order described avoids pattern work similar problems data streams general systems system considered element input data stream pushed data pushed way items appropriate order output stream observe given system certain derived book certain fixed order spine number pages equal number traffic control described book describe phases traffic signal controlled intersection intersection incoming outgoing lanes traffic including ends pedestrian bicycle lanes lanes motor vehicles represented placed spine book clockwise order junction paths intersection taken traffic incoming lane outgoing lane represented edges instance edge incoming outgoing lane traffic belong segment road representing turn segment segment turns allowed junction given edges represents collection paths traversed interference include pair edges cross edges placed single page book book describes partition paths non interfering book thickness fixed spine gives minimum number distinct phases needed signalling schedule includes possible traffic paths junction drawing book frequently applied network data standard drawing arc circular viewed book book applied construction simultaneous dimensional drawings arc linear places line draws edges line allowing edges drawn segments line drawing style corresponds book page line pages sides line originally introduced way studying crossing numbers page book drawn similar way allowing edges represented multiple line drawing book usual definition called book possible find edge crosses spine drawing style circular layout placed circle edges drawn inside outside circle placement edges circle instance straight line segments corresponds page book drawing placement inside outside circle corresponds page book drawing page drawings style important number crossings small way reducing visual drawing number crossings complete ratio number page page crossing number fixed number given combination crossing number methods reducing crossing complexity devised based careful order local page book fixed partition edges pages interpreted form given drawn way parts edges placed drawing way reflects page book find simultaneous given set find placement drawn straight edges book pages construct dimensional drawings particular wood construction book degree page low method dimensional grid low volume folding study molecules fold form structure standard form acid secondary structure described chain bases sequence drawn line collection arcs line describing structure structures actually complicated dimensional shape secondary structure exists described abstract structure page book behave simple way proposed called secondary structure certain takes form page book sequence drawn line drawn arcs line order form secondary structure maximum degree base participate arc addition links neighbors base sequence advantages include facts structures actually space matches known spine ordering known advance application testing existence secondary structure given straightforward problem assigning edges pages compatible way formulated instance problem testing circle edges describe crossings alternatively efficiently secondary structure exists input formed connecting bases cycle sequence order adding given edges characterization allows secondary structures recognized linear time instance testing connection secondary structures book proof certain problems secondary structure comparison structure secondary requires pages determining page number hard complexity theory book study complexity theory problem directed observed page directed solved space space complexity class time problems page directed requires power space book connected distinction complexity classes existence constant page number key step proving time simulation tape non machines tape non machines areas mathematics study applications book thickness abstract defined zero finite local ring making zero edge pair values product zero multi paper sequence studied book knots links showing described sequence symbols links demonstrated sequence local changes
Book embedding