
bhramana samvidaanam

kompinettoriyal ganithashaasthrathil, oro sheershakathinum chuttumulla oru graafinte arikukalude vrithaakrithiyilulla kramam vivarichukondu bhramana samvidaanangal (kompinettoriyal embeddingukal allengil kompinettoriyal maappukal ennum vilikkunnu) graafukalude embeddingukal ooriyantabil prathalangalil encode cheiyunnu.
oru bhramana samvidaanathinte kooduthal aupachaarikamaaya nirvachanathil jody kramamattangal ulppedunnu; atharam oru jody oru multigraph, oru uparithalam, uparithalathil multigraphinte rund sell embedding enniva nirnnayikkaan paryaapthamaan.
oro bhramana padhathiyum oru adanja ooriyantad uparithalathil (ooriyanteshan-samrakshikkunna topological thulyatha vare) bandhippicha multigraphinte savisheshamaaya 2-sell ulcherkkal nirvachikkunnu. neremarichu, oru ooriyantad clossed uparithalathil bandhippicha multigraph "jii" ulcherkkunnathu "jii" athinte adisthaana multigraph aayi ulla oru advitheeya bhramana samvidaanathe nirvachikkunnu. bhramana samvidaanangalum 2-sell-embadingukalum thammilulla eee adisthaana thulyatha 1890-kalil lother hephttar aadyamaayi eratta roopathil sthireekarikkukayum 1950-kalil ringle vyaapakamaayi upayogikkukayum cheythu. swathanthramaayi, edmunds sidhaanthathinte praadhamika roopam nalkukayum addehathinte padanatthinte vishadaamshangal cheruppakkaar janapriyamaakkukayum cheythu. multigraphukalilekkulla saamaanyavalkkaranam avatharippichathu groze alpert aanu.
rotteshan samvidaanangal raingold mattullavarum upayogikkunna "rotteshan maappukal" yumayi bandhappettirikkunnu, pakshe samaanamalla. (2002) graafukalude sig-sag uthppannam nirvachikkaan. oru bhramana samvidaanam oro sheershakathinum chuttumulla arikukalude vrithaakrithiyilulla kramam vyakthamaakkunnu, athesamayam oru bhramana bhoopadam oro sheershakathileyum arikukalude (vrithaakrithiyilullathallaa) kramappedutthal vyakthamaakkunnu. koodaathe, eathu graphinum rotteshan sisttangale nirvachikkaam, athesamayam reingold muthalaayava. avaye nirvachikkuka, bhramana bhoopadangal saadhaarana graafukalil parimithappedutthiyirikku.
aupachaarika nirvachanam.
aupachaarikamaayi, oru bhramana samvidaanathe oru jody (σ, θ) aayi nirvachikkunnu, avide σ, θ enniva ore graundu settu "b" ll pravartthikkunna kramappeduthalukalaanu, θ oru nischitha-pooyint-free involsionaanu, koodaathe σ, θ enniva srishtikkunna group <σ, θ> "b" ll parivarthanaparamaayi pravartthikkunnu.
oru ooriyantad uparithalathil bandhippichirikkunna multigraph "jii" yude 2-sell embeddingle ninnu oru bhramana samvidaanam labhikkunnathin, "bii" yill "jii" yude "dartukal" (allengil "flagukal" allengil "ardha-arikukal") adangiyirikkatte; athaayath, "jii" yude oro attatthinum nammal "bii" yude rund gadakangal roopappeduthunnu, oro attathe pooyintinum onnu. oru attathinu athinte rund end pooyintukaludeyum athe sheershakam ullappol polum, aaa attathinu nammal rund dartukal srishtikkunnu. bii" yude athe attathu ninnu roopappetta mattoru dart θ ("bii") aanennu nammal anuvadikkunnu; ithu vyakthamaayi nischitha poyintukalillaattha oru involsionaanu. σ ("b") gadikaaradishayil dart ayirikkatte, arikukalude chaakrika kramatthil "b" ll ninnu athe sheershakathilekku sambhavikkam, avide "gadikaaradisha" ennathu uparithalathinte ooriyanteshan nirvachikkunnu.
oru multigraph ooriyantabil ennaal ooriyantad allatha uparithalathil ulcherthittundengil, athu saadhaaranayaayi rund bhramana samvidaanangalumaayi poruthappedunnu, uparithalathinte rund ooriyanteshanukalil oronninum onnu. eee rund bhramana samvidaanangalkkum ore involsion θ undu, ennaal oru bhramana samvidaanathinte kramaparivarthanam σ ennathu mattu bhramana samvidaanathinte anubandha kramaparivartthanatthinte vipareethamaanu.
rotteshan sisttathil ninnu ulcherkkal veendedukkuka.
oru bhramana samvidaanathil ninnu oru multigraph veendedukkaan, σ-yude oro bhramanapathathinum oru sheershakavum θ-yude oro bhramanapathathinum oru arikum nammal roopappeduthunnu. eee rund bhramanapathangalkkum shoonyamallaatha intersection undengil oru attamulla insidantaanu oru vertex. athinaal, oru sheershakathile sambhavangalude ennam bhramanapathatthinte valuppamaan, oru attathu sambhavangalude ennam krithyamaayi rund aanu. oru bhramana samvidaanam oru connectucheytha multigraph "jii" yude 2-sell embeddingle ninnu uruthirinjathaanenkil, bhramana samvidaanathil ninnu uruthirinja graph isomorphic muthal "jii" vareyaanu.
oru bhramana samvidaanathil ninnu uruthirinja graph oru uparithalathilekku ulcherkkaan, σθ-yude oro bhramanapathathinum oru disk roopappeduthuka, koodaathe "e"-yumayi bandhapetta rund dartukal eee discukalumaayi bandhapetta rund bhramanapathangalil ulppedumbozhellaam "e" enna attathu rund discukal orumichu ottikkuka. σθ-yude bhramanapathangalumaayi bandhapetta discukalaaya rund cellukalulla deriwade multigraphinte 2-sell embedingaanu phalam. oro sheershakathinum chuttumulla arikukalude gadikaaradishayil kramam σ nalkunna gadikaaradishayil kramatthinu thulyamaaya reethiyil eee ulcherkkalinte uparithalam ooriyantad cheyyaan kazhiyum.
ulcherkkalinte uparithalathinte swabhaavam.
euler formula anusarichu rotteshan sisttam formula _ 1 (athaayath, adisthaana multigraph 2-sell ulchertha uparithalam) nirvachichirikkunna clossed ooriyantabil uparithalathinte "jii" januss namukk kandethaam. formula _ 2, formula _ 3, formula _ 4 enniva shradhikkuka.
ivide formula _ 6 ennathu kramaparivarthana formula _ 7-nte bhramanapathangalude koottatthe soochippikkunnu.

