Descripción de las prácticas relacionadas con el tema El Transistor

According to Theorem 14, the four minimal classes above the constant layer areR,E1_,R and_E1. Each of them contains polynomially manyn-vertex labelled graphs and hence the layer following the constant one is polynomial. In order to provide a global characterization of the polynomial layer we use the results and notation of Sections 2.2 and 2.3.

Theorem 15. For a hereditary class X, the following statements are equivalent: (1) log_|Xn|=⇥(logn);

(2) there exists a constant c such that every graph in Xn contains a similarity class of

size at least n c;

(3) complex degree andc-matching number are bounded for graphs inX; (4) none of the following classes is a subclass ofX:

B,_S,_Q,_M,_B,_S,_Q,_M.

Proof. (1) ) (4). It is not difficult to see that |Sn| = 2n n, |Bn| = 2n 1 and |Qn| = n2n 1 _{n(n+ 1)/2 + 1. Also, the number of n-vertex labelled graphs in M}

n is at least

bn/2c!. Therefore,X cannot contain any of the classesB,S,Q,M,B,S,Q,M.

(2)₎ (1). The number of labeled graphs on n vertices with a similarity class of size n k is equal to n_k 2(k+12 )+1 and hence we have that |logX_n|=⇥(logn).

Theorems 8 and 11 imply that (3) and (4) are equivalent. Therefore, it remains to show that (3) and (4) imply (2).

Let G be a graph in Xn. Since the c-matching number is bounded in G, we may

assume, without loss of generality, that µ(G)  k for a constant k. Therefore, n 2k vertices form an independent setI inG. Also, since the complex degree is bounded inG, there is a constantl such that every vertex v outside of I has at most lneighbours or at mostlnon-neighbours inI. By removing fromI eitherlneighbours orlnon-neighbours of v, for eachv62I, we transformI into a similarity class of sizen cwherec2k(l+1).

3.2.1 Coding of graphs in classes of polynomial growth

We will now describe another algorithm for coding graphs and provide a proof that it gives an asymptotically optimal coding for graphs in classes of polynomial growth.

Given a labelled graph Gon nvertices, define a new labelled graphG⇤ by choosing a similarity classK inGof greatest size and replacing the vertices inK with a single vertex z labelledn+ 1. The adjacencies of vertices to z are defined by x ₂G⇤_{\ {}z_} is adjacent toz if x is adjacent to each vertex in K, and x is non adjacent toz ifx is non adjacent to each vertex inK.

Now each vertex inGhas a label that can be described with a binary word of length at most logn. Let us denote by (1) the binary word consisting of the labels of the vertices in G_\K in increasing order. Let (2) ₌ c

n |K|+1(G⇤) and let (3) = 0 if K forms an independent set inGotherwise let (3)= 1 if K forms a clique in G.1 We now define

1

n(G) = (1) (2) (3), 1={ 1n : n= 1,2,3, . . .}.

Theorem 16. 1 is an asymptotically optimal coding for any hereditary class X with log_|Xn|=⇥(logn).

Proof. LetXbe a hereditary class satisfying|logXn|=⇥(logn). We know by Theorem 15

that there exists a constant c such that every graph in Xn contains a similarity class of

size at leastn c. PartitionX into setsX0, X1, . . . , Xc with Xi defined to be the set of graphs G₂X with the size of the largest similarity class contained in G precisely equal ton i for 0 i c. Define t to be the largest number such that the set Xt contains infinitely many graphs, i.e. there exists anN ₂Nsuch that for all n > N every graph in Xn has a similarity class of size at least n t. Hence forn > N we have that

max G2Xn| 1 n(G)|tlogn+ ✓ t+ 1 2 ◆ + 1.

We also know that|Xn| n_t and hence

log_|Xn| tlogn. Therefore, lim n!1 max G2Xn| 1 n(G)| log|Xn|  1.

However, each graph in Xn must have a unique coding so one must have length at least

log_|Xn|, hence the limit is equal to 1.

G 1 2 3 4 5 6 7 8 9

Example. LetGbe the graph (pictured above) with 9 vertices {1,2,3,4,5,6,7,8,9}and the following canonical code:

11010000 1010000 010000 11111 1111 000 00 0.

The largest size of a similarity class in G is four, and the vertices 6,7,8 and 9 form such a similarity class. Hence, the vertices inG\K are 1,2,3,4 and 5. This tells us that (1) _{= 0001 0010 0011 0100 0101. Now replacing the vertices 6,7,8 and 9 with one}

vertex labelled 10 gives usG⇤ (pictured below).

G⇤ 1 2 3 4 5 10

Looking at G⇤ gives us (2)= 11010 1010 010 11 1 and as the vertices 6,7,8 and 9 form an independent set inG we have (3)_{= 0. So,}

1

In this particular case, the method for coding a graph described in this section gives a binary word of equal length to the one given by the canonical code.