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# lect11 - Lecture 11 Matroid Independent System Consider a...

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Lecture 11 Matroid

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Independent System Consider a finite set S and a collection C of subsets of S. ( S , C ) is called an independent system if C A C B B A , i.e., it is hereditary. Each subset in C is called an independent set .
Matroid . } { : \ | | | | and , : property exchange the satisfies it if matroid a is ) , ( system t independen An C x B B A x B A C B A C S 5

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Matric Matroid ce. independen linearly for fact known well a is propery exchange The matroid. a is ) , ( Then . of subsets t independen linearly all of collection the and of vectors row of set the be Let . matrix a Consider C S S C M S M
Graphic Matroid . } { implies which ), , ( of components connected two connected edge an has Hence, . components connected | | | | has ) , ( and components connected | | | | has ) , ( that Note . with , consider property, exchange show To matroid. a is ) , ( Then . of subgraph acyclic an induces which of each sets edge all of collection the and Let ). , ( graph d) (undirecte a Consider C e B B V e A B V B V A V A V |B| |A| C B A C S M G C E S E V G G - - = = =

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Extension . in extension no has if in maximal is set t independen an , subset any For extension. no has it if maximal is set t independen An t. independen is } { and if set t independen an of extension an called is element An F I F F I S F x I I x I x
Maximal Independent Set ). ( ) ( , any for if only and if matroid a is ) , ( system t independen An } of subsets t independen maximal all over is | | {| min ) ( } | | {| max ) ( define , For ). , ( system t independen an Consider F v F u C F C S F A A F v C I I F F u S F C S = = = Theorem

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Proof . in extension an has so , of set t independen maximal a be cannot Hence, . | | | | size has of set t independen maxumal every
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lect11 - Lecture 11 Matroid Independent System Consider a...

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