Understanding Maximal Independent Sets in Graph Theory
Discover the concept of maximal independent sets in graph theory, where independence numbers, domination numbers, and claw-free graph theorems play a crucial role. Explore examples with paths, cycles, and claw-free graphs to deepen your understanding.
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Presentation Transcript
The maximal independent set between ?(?) and ?(?) Ma-Lian Chia
Independent Dominating => Dominating => Independent Set
Definition Independent Set For each pair of vertices ?,? in ? ?(?), we have ?? ?(?). The set of red vertices is an independent set. The set of green vertices is not an independent set.
Definition Independent Set Maximal Independent Set Maximal independent set is an independent set that is not a subset of any other independent set. The set of yellow vertices is a maximal independent set. The set of red vertices is not a maximal independent set.
Definition Independent Set Maximal Independent Set Independence Number ?(?) The cardinality of the largest independent vertex set Independent Domination Number ?(?) (or ??(?)) The minimum cardinality of maximal independent set ? ?6 = 3. ? ?6 = 2.
Our Question is: For each number ? between ?(?) and ? ? , is there a maximal independent set ? with ? = ??
The case of Path and Cycle ? ?? = ?/2 ,? ?? = ?/3 ? = ? = ?(?) ? = ? + 1 ? = ?(?)
The case of Path and Cycle ? ?? = ?/2 ,? ?? = ?/3 ? = ?(?) ? + 1 ? = ?(?)
Claw-free Graph Theorem If ? is a claw-free graph, then for each ?, ? ? ? ?(?), there is a maximal independent set ? with ? = ?. Suppose ? is a maximum independent set in ?, and ? is an arbitrary maximal independent set in ? with ? < |?|.
?:maximum independent set ?:a maximal independent set ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? is an induced subgraph with 2
? ? ? ? ?2 ?4 ?1 ?3 ?? 1 ?5 ?? ?? ? ? = ? ?2,?4, ,?? 1 ?1,?3, ,?? is a maximal independent set and ? = ? + 1.
?2?? Theorem For each ?, ? ?2 ?? = (? + 2)/2 ? ? ?2 ?? = ?, there is a maximal independent set ? with ? = ?. ?2 ?3 ?2 ?3
?2?? ? ?2 ?? = ? ? = ? 1 ? = ? 2
?3?? ? ?3 ?? = (3? + 8)/4 ,? 2 (mod 4) (3? + 4)/4 ,?? ??????. ? ?3 ?? = 3?/2 , Consider that ? = 4?,? ?3 ?? = 3? + 1
?3?? ? = 6 ? = 7
?3?? ? = 8 ? = 9
?3?? ? = 10 ? = 12
?3?? ? = 11
?3?? ? = 4?,? ?3 ?4? = 3? + 1
?3?? ? = 4? + 1,? ?3 ?4?+1 = 3? + 1
?3?? ? = 4? + 2,? ?3 ?4?+2 = 3? + 3
?3?? ? = 4? + 3,? ?3 ?4?+3 = 3? +3
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 5
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 6
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 7
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 8
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 9
Special Case - Complete ?-ary tree ? ? = 10,? ? = 5, |?| = 10
Special Case - Complete ?-ary tree ? ? = 5,? ? = 2, ? = 4 ?? ??? ?????.
