All Square Roots of a Knig-Egervry Graph Vadim E. Levit & Eugen - PowerPoint PPT Presentation

All Square Roots of a König-Egerváry Graph Vadim E. Levit & Eugen Mandrescu Ariel University, Israel & Holon Institute of Technology, Israel June 16-19, 2015 Algorithmic Graph Theory on the Adriatic Coast 2015 Koper, Slovenia Levit & Mandrescu (AU & HIT) Square Roots 19/06 1 / 55

Outline Some definitions : independent sets, matchings 1 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Square of graphs 3 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Square of graphs 3 Square-roots of graphs 4 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Square of graphs 3 Square-roots of graphs 4 Some old results 5 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Square of graphs 3 Square-roots of graphs 4 Some old results 5 Our findings ... 6 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Outline Some definitions : independent sets, matchings 1 König-Egerváry graphs 2 Square of graphs 3 Square-roots of graphs 4 Some old results 5 Our findings ... 6 Some open problems 7 Levit & Mandrescu (AU & HIT) Square Roots 19/06 2 / 55

Some definitions: independent sets � � x b � � � � � � G � � y a u c v Figure: G has α ( G ) = |{ a , b , c , y }| = 4. Definition An independent or a stable set is a set of pairwise non-adjacent vertices. The independence number or the stability number α ( G ) of G is the maximum cardinality of an independent set in G . Levit & Mandrescu (AU & HIT) Square Roots 19/06 3 / 55

Some definitions: independent sets � � x b � � � � � � G � � y a u c v Figure: G has α ( G ) = |{ a , b , c , y }| = 4. Definition An independent or a stable set is a set of pairwise non-adjacent vertices. The independence number or the stability number α ( G ) of G is the maximum cardinality of an independent set in G . Example { a } , { a , b } , { a , b , x } , { a , b , c , y } are independent sets of G . { a , b , c , x } , { a , b , c , y } are maximum independent sets, hence α ( G ) = 4. Levit & Mandrescu (AU & HIT) Square Roots 19/06 3 / 55

Some definitions: matchings and matching number Definition A matching in G is a set of non-incident edges. The matching number µ ( G ) of G is the maximum size of a matching in G . A matching covering all the vertices is called perfect. Levit & Mandrescu (AU & HIT) Square Roots 19/06 4 / 55

Some definitions: matchings and matching number Definition A matching in G is a set of non-incident edges. The matching number µ ( G ) of G is the maximum size of a matching in G . A matching covering all the vertices is called perfect. Example { a 1 a 2 } is a maximum matching in K 3 , hence µ ( K 3 ) = 1 { v 1 v 2 , v 3 v 4 } is maximum matching in C 5 , hence µ ( C 5 ) = 2 { t 1 t 2 , t 3 t 4 , t 5 t 5 } is maximum matching in G , hence µ ( G ) = 3 � � � � � � a 2 v 2 v 3 t 2 t 4 t 6 � � � � � � � � � � � � K 3 C 5 � � G v 5 � � � t 3 a 1 a 3 v 1 v 4 t 1 t 5 Figure: Only G has perfect matchings; e.g., M = { t 1 t 3 , t 2 t 4 , t 5 t 6 } . Levit & Mandrescu (AU & HIT) Square Roots 19/06 4 / 55

Some definitions: König-Egerváry graphs Remark �| V | / 2 � + 1 ≤ α ( G ) + µ ( G ) ≤ | V | hold for every graph G = ( V , E ) . Levit & Mandrescu (AU & HIT) Square Roots 19/06 5 / 55

Some definitions: König-Egerváry graphs Remark �| V | / 2 � + 1 ≤ α ( G ) + µ ( G ) ≤ | V | hold for every graph G = ( V , E ) . Definition (R. W. Deming (1979), F. Sterboul (1979)) G = ( V , E ) is a König—Egerváry graph if α ( G ) + µ ( G ) = | V | . Levit & Mandrescu (AU & HIT) Square Roots 19/06 5 / 55

Some definitions: König-Egerváry graphs Remark �| V | / 2 � + 1 ≤ α ( G ) + µ ( G ) ≤ | V | hold for every graph G = ( V , E ) . Definition (R. W. Deming (1979), F. Sterboul (1979)) G = ( V , E ) is a König—Egerváry graph if α ( G ) + µ ( G ) = | V | . � � � � v 2 v 4 x b � � � � � � � � � � G 1 � G 2 � � � y v 1 v 3 v 5 a u c v Figure: G 1 is a König—Egerváry graph, since α ( G 1 ) + µ ( G 1 ) = 7 = | V ( G 1 ) | , while G 2 is not a König—Egerváry graph, as α ( G 2 ) + µ ( G 2 ) = 4 < 5 = | V ( G 2 ) | . Theorem (D. König (1931), E. Egerváry (1931)) Each bipartite graph G = ( V , E ) satisfies α ( G ) + µ ( G ) = | V | . Levit & Mandrescu (AU & HIT) Square Roots 19/06 5 / 55

A characterization for König-Egerváry graphs Notation If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . If S ∈ Ind ( G ) and H = G − S , we write G = S ∗ H . Levit & Mandrescu (AU & HIT) Square Roots 19/06 6 / 55

A characterization for König-Egerváry graphs Notation If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . If S ∈ Ind ( G ) and H = G − S , we write G = S ∗ H . Theorem (Levit and Mandrescu, Discrete Math. 2003) For a graph G = ( V , E ) , the following properties are equivalent: (i) G is a König-Egerváry graph; (ii) G = S ∗ H , where S ∈ Ω ( G ) and | S | ≥ µ ( G ) = | V − S | ; (iii) G = S ∗ H , where S is an independent set with | S | ≥ | V − S | and ( S , V − S ) contains a matching of size | V − S | . � � � � � y g x f h � � � S = { x , y } ∈ Ω ( G ) � � � � � � � � G H c � � � µ ( G ) = 2 < | V − S | a v u b Figure: By above theorem, part (ii), only H is a König-Egerváry graph. Levit & Mandrescu (AU & HIT) Square Roots 19/06 6 / 55

Another characterization of König-Egerváry graphs Definition If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . Levit & Mandrescu (AU & HIT) Square Roots 19/06 7 / 55

Another characterization of König-Egerváry graphs Definition If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . Theorem (Levit and Mandrescu, Discrete Applied Math. 2013) For a graph G = ( V , E ) , the following properties are equivalent: (i) G is a König-Egerváry graph; Levit & Mandrescu (AU & HIT) Square Roots 19/06 7 / 55

Another characterization of König-Egerváry graphs Definition If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . Theorem (Levit and Mandrescu, Discrete Applied Math. 2013) For a graph G = ( V , E ) , the following properties are equivalent: (i) G is a König-Egerváry graph; (ii) each maximum matching is contained in ( S ∗ , V − S ∗ ) for some maximum independent set S ∗ ; Levit & Mandrescu (AU & HIT) Square Roots 19/06 7 / 55

Another characterization of König-Egerváry graphs Definition If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . Theorem (Levit and Mandrescu, Discrete Applied Math. 2013) For a graph G = ( V , E ) , the following properties are equivalent: (i) G is a König-Egerváry graph; (ii) each maximum matching is contained in ( S ∗ , V − S ∗ ) for some maximum independent set S ∗ ; (iii) each maximum matching is contained in ( S , V − S ) for every maximum independent set S . Levit & Mandrescu (AU & HIT) Square Roots 19/06 7 / 55

Another characterization of König-Egerváry graphs Definition If A ∩ B = ∅ in G = ( V , E ) , then ( A , B ) = { ab ∈ E : a ∈ A , b ∈ B } . Theorem (Levit and Mandrescu, Discrete Applied Math. 2013) For a graph G = ( V , E ) , the following properties are equivalent: (i) G is a König-Egerváry graph; (ii) each maximum matching is contained in ( S ∗ , V − S ∗ ) for some maximum independent set S ∗ ; (iii) each maximum matching is contained in ( S , V − S ) for every maximum independent set S . � � � � � y g S = { x , y } x f h � � � � � � � � � � � G H M = { ac , yb } c � � � a v u b Figure: M � ( S , V ( G ) − S ) , hence G is not a König-Egerváry graph. H is a König-Egerváry graph. Levit & Mandrescu (AU & HIT) Square Roots 19/06 7 / 55

� � � � � � � � � � � � � S � � � � � � � � � � � � � � G � � � � � � � � � � � � � � � � � � � � � � � � � � V − S A König-Egerváry graph G = S ∗ H Levit & Mandrescu (AU & HIT) Square Roots 19/06 8 / 55