Slither


A matching in G is a set M of edges of G, no two of which share a vertex. The matching M is a maximum matching in G if G has no matching with more edges then M. The size of any maximum matching of G is called the matching number of g denoted by alpha' (G).

The matching M is said to be saturate vertex v if v is an end of an edge of M, otherwise v is said to be M-unsaturated. The vertex v is universal in G if every maximum matching of G saturates v, so lambda' (G/v)=alpha' (G)-1. (And conversely, if v is not universal in G, then alpha' (G/v)=alpha' (G).

 

The Slither Theorem

I.) If v0 is universal in G, then the first player can force a win in Slither as follows. Let M be any maximum matching in G. The first players winning strategy is to always pick edges from M.

II.)If v0 is not universal in G, then the second player can force a win in Slither as follows. Let M be any maximum matching in G not saturating v0. The second playerís winning strategy is to always pick edges from M.

  1. What if player 1 canít follow the strategy?

 

M' is a maximum matching not saturating v0.

IV.)What if player 2 canít follow the strategy?

M' is a layer matching.

An M-alternating path is a path whose edges are alternately in M and not in M. An M-augmenting path is an M-alternating path with two M-unsaturated ends.

True Fact

If there is an M-augmenting path,

then M is not a maximum matching.

Let M and N be matching in G, let H be the graph with the same vertex set as G, and edges M (union) N. Then the components of H are each, one of the following six types.

Claude Berge's Theorem

Let M be a matching in G that is not maximum. Then G has an M-augmenting path.

Proof: Let N be a larger matching. Then H must have a type 5 component, which in an M-augmenting path.

Dom de Cain's Theorem

Let e=uv be an edge of G. Then:

Either

1.u is universal in G

2.v is universal in G

3.e is part of an odd cycle

Proof: If #1 is not true let M be a maximum matching not saturating u. Then M must saturate v, otherwise M u{e} would be a larger matching. Similarly, if #2 is not true then there is a maximum matching N not saturating v, which must saturate u:

 

For each vertex x, let Tx be the component of H containing x. Then Tu and Tv are paths of type 3. If Tu does not equal Tv , preform a switcheroony on Tv:

 

This produces a larger matching M' u{e}, a contradiction. Thus Tu = Tv, and Tu u{e} is an odd cycle, so #3 Holds.

Theorem

Let G be a connected graph on n vertices, non of which are universal. Then u is odd, and alpha'(G)=1/2(n-1)