Game f -matching in Graphs Douglas B. West Zhejiang Normal - PowerPoint PPT Presentation

The 1 -Matching Game Write ν k ( G ) and ˆ ν k ( G ) when f (  ) = k for all  . Prior results (Cranston–Kinnersley–O–West [2012]): Thm. | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 for every graph G . Thm. If H is an induced subgraph of G , then ν 1 ( H ) ≤ ν 1 ( G ) and ˆ ν 1 ( H ) ≤ ˆ ν 1 ( G ) . Thm. All positive pairs ( , j ) except ( 1 , 2 ) occur. Thm. ν 1 ( G ) ≥ 2 3 m 1 ( G ) , which is sharp. � 3 � Thm. ν 1 ( G ) ≤ min 2 m 1 ( G ) , m 1 ( G ) , which is sharp. Thm. ν 1 ( G ) ≥ 3 4 m 1 ( G ) when G is a forest (sharp). Also ˆ ν 1 ( G ) ≤ ν 1 ( G ) for forests. Which extend to f -matching or at least 2 -matching?

The Easy Lower Bound Generalizes Thm. ν f ( G ) ≥ 2 3 m f ( G ) for every graph G .

The Easy Lower Bound Generalizes Thm. ν f ( G ) ≥ 2 3 m f ( G ) for every graph G . Pf. Consider a round of play.

The Easy Lower Bound Generalizes Thm. ν f ( G ) ≥ 2 3 m f ( G ) for every graph G . Pf. Consider a round of play. Max plays an edge  of a maximum f -matching M . Let G ′ = G −  and f ′ = { ,  } -reduction of f . Note m f ′ ( G ′ ) ≥ m f ( G ) − 1 .

The Easy Lower Bound Generalizes Thm. ν f ( G ) ≥ 2 3 m f ( G ) for every graph G . Pf. Consider a round of play. Max plays an edge  of a maximum f -matching M . Let G ′ = G −  and f ′ = { ,  } -reduction of f . Note m f ′ ( G ′ ) ≥ m f ( G ) − 1 . Min plays some edge y . Let G ′′ = G ′ − y and f ′′ = { , y } -reduction of f ′ . Deleting edges from M ′ at  and y leaves f ′′ -matching. Thus m f ′′ ( G ′′ ) ≥ m f ′ ( G ′ ) − 2 . − 1 − 1 • • • • • • • •

The Easy Lower Bound Generalizes Thm. ν f ( G ) ≥ 2 3 m f ( G ) for every graph G . Pf. Consider a round of play. Max plays an edge  of a maximum f -matching M . Let G ′ = G −  and f ′ = { ,  } -reduction of f . Note m f ′ ( G ′ ) ≥ m f ( G ) − 1 . Min plays some edge y . Let G ′′ = G ′ − y and f ′′ = { , y } -reduction of f ′ . Deleting edges from M ′ at  and y leaves f ′′ -matching. Thus m f ′′ ( G ′′ ) ≥ m f ′ ( G ′ ) − 2 . − 1 − 1 • • • • • • • • ∴ Each round plays 2 edges and reduces max size of achievable subgraph by at most 3 .

Sharpness Ex. Let G = K n K n and f (  ) = k for all  ∈ V ( G ) . m k ( G ) = 1 ν k ( G ) = 2 Note m k ( G ) = kn , 2 kn , 3 kn . k • • • • k • • • • k S T • • k • • • • k • • k

Sharpness Ex. Let G = K n K n and f (  ) = k for all  ∈ V ( G ) . m k ( G ) = 1 ν k ( G ) = 2 Note m k ( G ) = kn , 2 kn , 3 kn . k • • • • k • • • • k S T • • k • • • • k • • k Min can reduce capacity in T by 2 with each move.

Sharpness Ex. Let G = K n K n and f (  ) = k for all  ∈ V ( G ) . m k ( G ) = 1 ν k ( G ) = 2 Note m k ( G ) = kn , 2 kn , 3 kn . k • • • • k • • • • k S T • • k • • • • k • • k Min can reduce capacity in T by 2 with each move. • Always m f ( G ) ≥ 1 2 m f ( G ) .

Sharpness Ex. Let G = K n K n and f (  ) = k for all  ∈ V ( G ) . m k ( G ) = 1 ν k ( G ) = 2 Note m k ( G ) = kn , 2 kn , 3 kn . k • • • • k • • • • k S T • • k • • • • k • • k Min can reduce capacity in T by 2 with each move. • Always m f ( G ) ≥ 1 2 m f ( G ) . Uses that an f -matching M is a maximum f -matching if and only if G has no M -augmenting trail.

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching.

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality.

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 . Max plays some edge, reducing M -degree by ≥ 1 .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 . Max plays some edge, reducing M -degree by ≥ 1 . Reducing M -degree by 2 kills one or two edges of M .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 . Max plays some edge, reducing M -degree by ≥ 1 . Reducing M -degree by 2 kills one or two edges of M . Min plays in M for r rounds, s killing 3 edges: 2 r + s ≥| M | . These 2 r moves reduce M -degree by at least 3 r + s .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 . Max plays some edge, reducing M -degree by ≥ 1 . Reducing M -degree by 2 kills one or two edges of M . Min plays in M for r rounds, s killing 3 edges: 2 r + s ≥| M | . These 2 r moves reduce M -degree by at least 3 r + s . Remaining M -degree (and moves) are ≤ 2 | M | − 3 r − s .

The Easy Upper Bound Generalizes Thm. ν f ( G ) ≤ 3 2 m f ( G ) for every graph G . Pf. Let M be a smallest maximal f -matching. Vertices with f (  ) > d M (  ) are nonadjacent or joined by an edge of M , by maximality. Min plays in M when possible, reducing M -degree by 2 . Max plays some edge, reducing M -degree by ≥ 1 . Reducing M -degree by 2 kills one or two edges of M . Min plays in M for r rounds, s killing 3 edges: 2 r + s ≥| M | . These 2 r moves reduce M -degree by at least 3 r + s . Remaining M -degree (and moves) are ≤ 2 | M | − 3 r − s . Hence ν f ( G ) ≤ 2 | M | − r − s ≤ 3 2 | M | , since 2 r + s ≥ | M | implies r + s ≥ | M | / 2 .

Sharpness Ex. Let G consist of t copies of P 4 with edge-multiplicity k (with kt even) and f (  ) = k for all  . Here m f ( G ) = kt and ν f ( G ) = 3 kt/ 2 . • • • • • • • • • • • • • • • •

Sharpness Ex. Let G consist of t copies of P 4 with edge-multiplicity k (with kt even) and f (  ) = k for all  . Here m f ( G ) = kt and ν f ( G ) = 3 kt/ 2 . • • • • • • • • • • • • • • • • Ex. Let G consist of K k + 1 with k pendant edges at each � k + 1 � . ν f ( G ) = 3 vertex, and f (  ) = k for all  ; 2 2 • • • •• • • • • • • •• • • •

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move.

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Key: If f ≡ 1 , then ν f ′ ( G −  ) = ν 1 ( G − { ,  } ) .

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Key: If f ≡ 1 , then ν f ′ ( G −  ) = ν 1 ( G − { ,  } ) . Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V ( G ) , (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) .

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Key: If f ≡ 1 , then ν f ′ ( G −  ) = ν 1 ( G − { ,  } ) . Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V ( G ) , (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G .

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Key: If f ≡ 1 , then ν f ′ ( G −  ) = ν 1 ( G − { ,  } ) . Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V ( G ) , (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G . Let  and y be optimal starts for Max and Min on G . ν 1 ( G ) = 1 + ˆ ν 1 ( G −  −  ) ≤ 1 + ˆ ν 1 ( G −  ) ≤ 1 + ˆ ν 1 ( G ) . ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≤ 1 + ν 1 ( G − y ) ≤ 1 + ν 1 ( G ) . ˆ

Always | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 Def. S -reduction of f reduces capacity by 1 for  ∈ S . Prop. If  ∈ E ( G ) and f ′ is { ,  } -reduction of f , then ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) and ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Key: If f ≡ 1 , then ν f ′ ( G −  ) = ν 1 ( G − { ,  } ) . Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V ( G ) , (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G . Let  and y be optimal starts for Max and Min on G . ν 1 ( G ) = 1 + ˆ ν 1 ( G −  −  ) ≤ 1 + ˆ ν 1 ( G −  ) ≤ 1 + ˆ ν 1 ( G ) . ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≤ 1 + ν 1 ( G − y ) ≤ 1 + ν 1 ( G ) . ˆ Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G . Let H = G −  . Let y be optimal start for Max on H . ν 1 ( G ) ≥ 1 + ˆ ν 1 ( G −  − y ) ≥ 1 + ˆ ν 1 ( H −  − y ) = ν 1 ( H ) .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G . Let H = G −  . Let y be optimal start for Max on H . ν 1 ( G ) ≥ 1 + ˆ ν 1 ( G −  − y ) ≥ 1 + ˆ ν 1 ( H −  − y ) = ν 1 ( H ) . ∈ { , y } , then Let y be optimal start for Min on G . If  / ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≥ 1 + ν 1 ( H −  − y ) ≥ ˆ ˆ ν 1 ( H ) .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G . Let H = G −  . Let y be optimal start for Max on H . ν 1 ( G ) ≥ 1 + ˆ ν 1 ( G −  − y ) ≥ 1 + ˆ ν 1 ( H −  − y ) = ν 1 ( H ) . ∈ { , y } , then Let y be optimal start for Min on G . If  / ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≥ 1 + ν 1 ( H −  − y ) ≥ ˆ ˆ ν 1 ( H ) . If  =  and z ∈ N H ( y ) , then using first move yz in H , ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≥ 1 + ν 1 ( G −  − y − z ) ≥ ˆ ˆ ν 1 ( H ) .

Completion of proof Prop. ν f ( G ) ≥ 1 + ˆ ν f ′ ( G −  ) & ˆ ν f ( G ) ≤ 1 + ν f ′ ( G −  ) . Equality ⇔  is optimal first move. Thm. (1) | ν 1 ( G ) − ˆ ν 1 ( G ) | ≤ 1 , and (2) ν 1 ( G ) ≥ ν 1 ( G −  ) and ˆ ν 1 ( G ) ≥ ˆ ν 1 ( G −  ) . Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G . Let H = G −  . Let y be optimal start for Max on H . ν 1 ( G ) ≥ 1 + ˆ ν 1 ( G −  − y ) ≥ 1 + ˆ ν 1 ( H −  − y ) = ν 1 ( H ) . ∈ { , y } , then Let y be optimal start for Min on G . If  / ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≥ 1 + ν 1 ( H −  − y ) ≥ ˆ ˆ ν 1 ( H ) . If  =  and z ∈ N H ( y ) , then using first move yz in H , ν 1 ( G ) = 1 + ν 1 ( G −  − y ) ≥ 1 + ν 1 ( G −  − y − z ) ≥ ˆ ˆ ν 1 ( H ) . If  =  and d H ( y ) = 0 , then ν 1 ( G ) = 1 + ν 1 ( G −  − y ) = 1 + ν 1 ( H − y ) = 1 + ν 1 ( H ) ≥ ˆ ˆ ν 1 ( H ) .

Open Problem Conj. (1) | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 for general G and f .

Open Problem Conj. (1) | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 for general G and f . Possible added monotonicity statements for induction:

Open Problem Conj. (1) | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 for general G and f . Possible added monotonicity statements for induction: • (2) If f (  ) ≥ 1 , and h is the {  } -reduction of f , then ν f ( G ) ≥ ν h ( G ) and ˆ ν f ( G ) ≥ ˆ ν h ( G ) .

Open Problem Conj. (1) | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 for general G and f . Possible added monotonicity statements for induction: • (2) If f (  ) ≥ 1 , and h is the {  } -reduction of f , then ν f ( G ) ≥ ν h ( G ) and ˆ ν f ( G ) ≥ ˆ ν h ( G ) . • (2) If f (  ) , f (  ) ≥ 1 , and f ′ is the { ,  } -reduction of f , then ν f ( G ) ≥ ν f ′ ( G −  ) and ˆ ν f ( G ) ≥ ˆ ν f ′ ( G −  ) .

Open Problem Conj. (1) | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 for general G and f . Possible added monotonicity statements for induction: • (2) If f (  ) ≥ 1 , and h is the {  } -reduction of f , then ν f ( G ) ≥ ν h ( G ) and ˆ ν f ( G ) ≥ ˆ ν h ( G ) . • (2) If f (  ) , f (  ) ≥ 1 , and f ′ is the { ,  } -reduction of f , then ν f ( G ) ≥ ν f ′ ( G −  ) and ˆ ν f ( G ) ≥ ˆ ν f ′ ( G −  ) . Parts of the argument for 1 -matching generalize, but it seems harder to use these to prove (1).

An Easy Directed Version Idea: Impose capacity f (  ) only on the outdegree of  .

An Easy Directed Version Idea: Impose capacity f (  ) only on the outdegree of  . Given undirected G , players Max and Min alternately select and orient an edge for the subgraph H so that always d + H (  ) ≤ f (  ) at each  ∈ V ( G ) .

An Easy Directed Version Idea: Impose capacity f (  ) only on the outdegree of  . Given undirected G , players Max and Min alternately select and orient an edge for the subgraph H so that always d + H (  ) ≤ f (  ) at each  ∈ V ( G ) . They aim to maximize and minimize the final | E ( H ) | , respectively.

An Easy Directed Version Idea: Impose capacity f (  ) only on the outdegree of  . Given undirected G , players Max and Min alternately select and orient an edge for the subgraph H so that always d + H (  ) ≤ f (  ) at each  ∈ V ( G ) . They aim to maximize and minimize the final | E ( H ) | , respectively. Let µ f ( G ) and ˆ µ f ( G ) denote the number of edges selected under optimal play in the Max-start and Min-start games, respectively.

An Easy Directed Version Idea: Impose capacity f (  ) only on the outdegree of  . Given undirected G , players Max and Min alternately select and orient an edge for the subgraph H so that always d + H (  ) ≤ f (  ) at each  ∈ V ( G ) . They aim to maximize and minimize the final | E ( H ) | , respectively. Let µ f ( G ) and ˆ µ f ( G ) denote the number of edges selected under optimal play in the Max-start and Min-start games, respectively. Thm. For every graph G and capacity function f on G , | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) . Let Y consist of f (  ) copies of  for all  ∈ V ( G ) .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) . Let Y consist of f (  ) copies of  for all  ∈ V ( G ) . Make  ∈ X adjacent in G ′ to all copies in Y of  and  .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) . Let Y consist of f (  ) copies of  for all  ∈ V ( G ) . Make  ∈ X adjacent in G ′ to all copies in Y of  and  . Since | ν 1 ( G ′ ) − ˆ ν 1 ( G ′ ) | ≤ 1 , it suffices to show µ f ( G ) = ν 1 ( G ′ ) and ˆ ν 1 ( G ′ ) . µ f ( G ) = ˆ

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) . Let Y consist of f (  ) copies of  for all  ∈ V ( G ) . Make  ∈ X adjacent in G ′ to all copies in Y of  and  . Since | ν 1 ( G ′ ) − ˆ ν 1 ( G ′ ) | ≤ 1 , it suffices to show µ f ( G ) = ν 1 ( G ′ ) and ˆ ν 1 ( G ′ ) . µ f ( G ) = ˆ Selecting e oriented away from  in the directed f -matching game on G corresponds to picking edge e ′ in the 1 -matching game on G ′ for some copy  ′ of  .

Transformation Argument Thm. | µ f ( G ) − ˆ µ f ( G ) | ≤ 1 for all G and capacity f . Pf. Build auxiliary ( X, Y ) -bigraph G ′ . Let X = E ( G ) . Let Y consist of f (  ) copies of  for all  ∈ V ( G ) . Make  ∈ X adjacent in G ′ to all copies in Y of  and  . Since | ν 1 ( G ′ ) − ˆ ν 1 ( G ′ ) | ≤ 1 , it suffices to show µ f ( G ) = ν 1 ( G ′ ) and ˆ ν 1 ( G ′ ) . µ f ( G ) = ˆ Selecting e oriented away from  in the directed f -matching game on G corresponds to picking edge e ′ in the 1 -matching game on G ′ for some copy  ′ of  . Each e ∈ E ( G ) = X is selected at most once. Each  ∈ V ( G ) is made tail (matched) ≤ f (  ) times.

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 .

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair?

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair? Obs. The 2 -matching game on K n is near-fair, since maximal 2 -matchings have n − 1 or n edges.

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair? Obs. The 2 -matching game on K n is near-fair, since maximal 2 -matchings have n − 1 or n edges. Thm. (Carraher–Kinnersley–Reiniger–West [2013+]) For n ≥ 5 (and n � = 7 ), always ν 2 ( K n ) � = ˆ ν 2 ( K n ) , with Player 1 “winning” for even n and Player 2 “winning” for odd n .

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair? Obs. The 2 -matching game on K n is near-fair, since maximal 2 -matchings have n − 1 or n edges. Thm. (Carraher–Kinnersley–Reiniger–West [2013+]) For n ≥ 5 (and n � = 7 ), always ν 2 ( K n ) � = ˆ ν 2 ( K n ) , with Player 1 “winning” for even n and Player 2 “winning” for odd n . � Thm. (Wise–West) If G = K n  with all n  having the same parity (and not in {1 , 2 , 3 , 4 , 7} ), then G is � ( n  − 1 ) and near-fair. Also, the values are � 1 + ( n  − 1 ) , with Player 2 winning unless G consists of an odd number of even-order components.

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair? Obs. The 2 -matching game on K n is near-fair, since maximal 2 -matchings have n − 1 or n edges. Thm. (Carraher–Kinnersley–Reiniger–West [2013+]) For n ≥ 5 (and n � = 7 ), always ν 2 ( K n ) � = ˆ ν 2 ( K n ) , with Player 1 “winning” for even n and Player 2 “winning” for odd n . � Thm. (Wise–West) If G = K n  with all n  having the same parity (and not in {1 , 2 , 3 , 4 , 7} ), then G is � ( n  − 1 ) and near-fair. Also, the values are � 1 + ( n  − 1 ) , with Player 2 winning unless G consists of an odd number of even-order components. Uses edge-transitivity of K n .

Another Question Def. G with capacity f is near-fair if | ν f ( G ) − ˆ ν f ( G ) | ≤ 1 . Ques. When G and H with capacities are near-fair, under what conditions must G + H be near-fair? Obs. The 2 -matching game on K n is near-fair, since maximal 2 -matchings have n − 1 or n edges. Thm. (Carraher–Kinnersley–Reiniger–West [2013+]) For n ≥ 5 (and n � = 7 ), always ν 2 ( K n ) � = ˆ ν 2 ( K n ) , with Player 1 “winning” for even n and Player 2 “winning” for odd n . � Thm. (Wise–West) If G = K n  with all n  having the same parity (and not in {1 , 2 , 3 , 4 , 7} ), then G is � ( n  − 1 ) and near-fair. Also, the values are � 1 + ( n  − 1 ) , with Player 2 winning unless G consists of an odd number of even-order components. Uses edge-transitivity of K n . Unions of components with different parities are hard to handle.

Back to Saturation Games The k -matching game on G is the same as the K 1 ,k + 1 -saturation game on G .

Back to Saturation Games The k -matching game on G is the same as the K 1 ,k + 1 -saturation game on G . Some results on saturation games: ( G ; F ) st ( G ; F ) s = st g ( G ; F ) ex ( G ; F ) Ω ( n lg n ) ≤ s ≤ n 2 / 5? n 2 / 4 ( K n , K 3 ) n − 1 ( K n ; P 4 ) n/ 2 ≈ 4 n/ 5 n or n − 1  n n even  m + n − 1 � = � ( K m,n ; P 4 ) n − 2 mn odd st g ( G ; F )  2 else m n 3 / 2 + O ( n 4 / 3 ) s > Ω ( n 13 / 12 ) ( K n,n ; C 4 ) n − 1 Füredi–Reimer–Seress [1991] for lower bound on st ( K n ; K 3 ) . Carraher–Kinnersley–Reiniger–West [2013+] for others.

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges.

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges. Lem. Let G be C 4 -saturated in K n,n with parts X and Y . If ∃ S ⊆ V ( G ) and c, d such that | S ∩ X | , | S ∩ Y | ≥ cn and | N G (  ) ∩ S | ≤ d � n for all  , then | E ( G ) | ≥ n 13 / 12 , 2 ( c 2 2 d 2 ) 2 / 3 , c 2 where  = min{ 1 2 d } . • X S ∩ X ≥ cn ≤ d � n Y S ∩ Y ≥ cn •

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges. Lem. Let G be C 4 -saturated in K n,n with parts X and Y . If ∃ S ⊆ V ( G ) and c, d such that | S ∩ X | , | S ∩ Y | ≥ cn and | N G (  ) ∩ S | ≤ d � n for all  , then | E ( G ) | ≥ n 13 / 12 , 2 ( c 2 2 d 2 ) 2 / 3 , c 2 where  = min{ 1 2 d } . • • X S ∩ X ≥ cn  Y S ∩ Y ≥ cn y • • ≥ c 2 n 2 − cdn 3 / 2 such S -paths. C 4 -sat ⇒

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges. Lem. Let G be C 4 -saturated in K n,n with parts X and Y . If ∃ S ⊆ V ( G ) and c, d such that | S ∩ X | , | S ∩ Y | ≥ cn and | N G (  ) ∩ S | ≤ d � n for all  , then | E ( G ) | ≥ n 13 / 12 , 2 ( c 2 2 d 2 ) 2 / 3 , c 2 where  = min{ 1 2 d } . • • X S ∩ X ≥ cn  Y S ∩ Y ≥ cn y • • ≥ c 2 n 2 − cdn 3 / 2 such S -paths. C 4 -sat ⇒ In half, central edge has endpt of degree < n 5 / 12 .

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges. Lem. Let G be C 4 -saturated in K n,n with parts X and Y . If ∃ S ⊆ V ( G ) and c, d such that | S ∩ X | , | S ∩ Y | ≥ cn and | N G (  ) ∩ S | ≤ d � n for all  , then | E ( G ) | ≥ n 13 / 12 , 2 ( c 2 2 d 2 ) 2 / 3 , c 2 where  = min{ 1 2 d } . • • X S ∩ X ≥ cn  Y S ∩ Y ≥ cn y • • ≥ c 2 n 2 − cdn 3 / 2 such S -paths. C 4 -sat ⇒ In half, central edge has endpt of degree < n 5 / 12 . Each such is central edge for at most dn 11 / 12 S -paths.

Sketch of Lower Bound Idea: Max constructs a subgraph of K n,n whose C 4 -saturated supergraphs have Ω ( n 13 / 12 ) edges. Lem. Let G be C 4 -saturated in K n,n with parts X and Y . If ∃ S ⊆ V ( G ) and c, d such that | S ∩ X | , | S ∩ Y | ≥ cn and | N G (  ) ∩ S | ≤ d � n for all  , then | E ( G ) | ≥ n 13 / 12 , 2 ( c 2 2 d 2 ) 2 / 3 , c 2 where  = min{ 1 2 d } . • • X S ∩ X ≥ cn  Y S ∩ Y ≥ cn y • • ≥ c 2 n 2 − cdn 3 / 2 such S -paths. C 4 -sat ⇒ In half, central edge has endpt of degree < n 5 / 12 . Each such is central edge for at most dn 11 / 12 S -paths. ∴ at least c 2 2 d n 13 / 12 such edges.

Max Strategy 1 10 . 4 n 13 / 12 . Thm. st g ( K n,n ; C 4 ) ≥

Max Strategy 1 10 . 4 n 13 / 12 . Thm. st g ( K n,n ; C 4 ) ≥ Pf. In first 2 n/ 3 moves, Max gives degree k to k �� � specified vertices in each part, where k = n/ 3 − 1 , by joining them to isolated vertices on the other side. X • • • • • • • • • • • • • • • • Y

Max Strategy 1 10 . 4 n 13 / 12 . Thm. st g ( K n,n ; C 4 ) ≥ Pf. In first 2 n/ 3 moves, Max gives degree k to k �� � specified vertices in each part, where k = n/ 3 − 1 , by joining them to isolated vertices on the other side. X • • • • • • • • • • • • • • • • Y Since G has no 4 -cycle, each vertex has at most one leaf neighbor in each star.

Max Strategy 1 10 . 4 n 13 / 12 . Thm. st g ( K n,n ; C 4 ) ≥ Pf. In first 2 n/ 3 moves, Max gives degree k to k �� � specified vertices in each part, where k = n/ 3 − 1 , by joining them to isolated vertices on the other side. X • • • • • • • • • • • • • • • • Y Since G has no 4 -cycle, each vertex has at most one leaf neighbor in each star. � ∴ With c ≈ 1 / 3 and d = 1 / 3 , the conditions of the lemma hold.

One More Open Problem Ques. For 3 -regular connected n -vertex graphs with perfect matchings, how small can ν 1 ( G ) be?

One More Open Problem Ques. For 3 -regular connected n -vertex graphs with perfect matchings, how small can ν 1 ( G ) be? Thm. For 3 -regular connected n -vertex graphs with perfect matchings, n/ 3 ≤ min ν 1 ( G ) ≤ 3 n/ 7 .

One More Open Problem Ques. For 3 -regular connected n -vertex graphs with perfect matchings, how small can ν 1 ( G ) be? Thm. For 3 -regular connected n -vertex graphs with perfect matchings, n/ 3 ≤ min ν 1 ( G ) ≤ 3 n/ 7 . • • • • • • • • • • • • • • • • • • • • • • • • • • • •