Chapter 4 Permutations and permutation graphs
5.4 Decidability of the Bell number
As we mentioned in the introduction, every class below the Bell number can be characterised by a finite set of forbidden induced subgraphs. Therefore all classes for which the set of minimal forbidden induced subgraphs is infinite have speed above the Bell number. For classes defined by finitely many forbidden induced subgraphs, the problem of deciding whether their speed is above the Bell number is more complicated and decidability of this problem has been an open question. In this section, we employ our characterisation of minimal classes above the Bell number to answer this question positively.
Our main goal is to provide an algorithm that decides for an input consisting of a finite number of graphsF1, . . . , Fn whether the speed of X = Free(F1, . . . , Fn)
is above the Bell number. That is, we are interested in the following problem.
Problem 5.4.1.
Input: A finite set of graphsF ={F1, F2, . . . , Fn}
Output: Yes, if the speed ofX = Free(F) is above the Bell number; no otherwise. Our algorithm, following the characterisation of minimal classes above the Bell number, distinguishes two cases depending on whether the distinguishing num- ber kX is finite or infinite. First we show how to discriminate between these two cases.
Problem 5.4.2.
Input: A finite set of graphsF ={F1, F2, . . . , Fn}
Output: Yes, if kX =∞ forX = Free(F); no otherwise.
Theorem 5.4.1. There is a polynomial-time algorithm that solves Problem 5.4.2.
Proof. By Theorem 5.3.1, kX = ∞ if and only if X contains one of the thirteen
minimal classes listed there. By Theorem 5.3.3, each of the minimal classes is defined by finitely many forbidden induced subgraphs; thus membership can be tested in polynomial time. Then the answer to Problem 5.4.2 is no if and only if each of the minimal classes given by Theorem 5.3.1 contains at least one of the graphs in F, which can also be tested in polynomial time.
By Corollary 5.3.1, the minimal hereditary classes with finite distinguishing number with speed above the Bell number can be described as P(w, H) with an almost periodic infinite word w. Here we give a more precise characterisation for classes defined by finitely many forbidden induced subgraphs.
Definition 5.4.1. Letw=w1w2. . .be an infinite word over a finite alphabetA. If
there exists somepsuch thatwi =wi+p for alli∈N, we call the wordwperiodicand
the number p its period. If, moreover, for some period p the letters w1, w2, . . . , wp
are all distinct, we call the wordw cyclic.
If w is a finite word, then w0 = (w)∞ is the periodic word obtained by concatenating infinitely many copies of the wordw; thuswi0 =wkfork=imod|w0|.
A class X of graphs is called a periodic class (cyclic class, respectively) if there exists a graphHwith loops allowed and a periodic (cyclic, respectively) wordw such thatX =P(w, H).
Definition 5.4.2. Let A = {1,2, . . . , `} be a finite alphabet, H a graph on A with loops allowed, andM a positive integer. Define a graph SH,M with vertex set
V(SH,M) = A× {1,2, . . . , M} and an edge between (a, j) and (b, k) if and only if
one of the following holds:
• ab∈E(H) and either |a−b| 6= 1 or j6=k;
• ab /∈E(H) and |a−b|= 1 and j=k. The graphSH,M is called an (`, M)-strip.
Notice that a strip can be viewed as the graph obtained from the union of M disjoint paths (1, j)−(2, j)− · · · −(`, j) for j ∈ {1,2, . . . , M} by swapping edges with non-edges between vertices (a, j) and (b, k) if ab∈E(H).
Theorem 5.4.2. Let X = Free(F1, F2, . . . , Fn) with kX finite. Then the following conditions are equivalent:
(a) The speed ofX is above the Bell number.
(b) X contains a periodic class.
(c) For every p∈N,X contains a cyclic class with period at least p.
(d) There exists a cyclic wordwand a graphHon the alphabet ofwsuch thatX con-
tains the `-factor Gw,H(1,2, . . . ,2`m) with `=|V(H)|and m = max
|Fi|:i∈
{1,2, . . . , n} .
(e) For any positive integers`, m, the class X contains an (`, m)-strip.
Proof. (a) ⇒ (b): From Theorem 5.3.5 we know that X contains a class P(w, H)
with some almost periodic word w = w1w2. . . and a finite graph H with loops
allowed. Let m = max{|F1|,|F2|, . . . ,|Fn|} and let a = w1w2. . . wm be the word
consisting of the first m letters of the infinite word w. Since w is almost periodic, the factora appears in w infinitely often. In particular, there ism0 > msuch that wm0+1wm0+2. . . wm0+m =a. Define bto be the word between the two a’s in w, i.e.,
We claim that X contains the periodic class P(w0, H) with w0 = (ab)∞. For the sake of contradiction, suppose that X does not contain P(w0, H). Then for some i ∈ {1,2, . . . , n}, we have Fi ∈ P(w0, H). So Fi ∼= Gw0,H(u1, u2, . . . , uk)
for some u1 < u2 < · · · < uk. Let U = {u1, u2, . . . , uk}. We are now looking
for a monotonically increasing function f : U → N with these two properties:
firstly, wf(u) = w0u for any u ∈ U; secondly, f(u) −f(u0) = 1 if and only if u−u0 = 1. If we can establish the existence of such a function, we will then have Fi ∼=Gw0,H(u1, u2, . . . , uk)∼=Gw,H(f(u1), f(u2), . . . , f(uk))∈ P(w, H) ⊆ X, a
contradiction.
To construct such a functionf, consider a maximal block{uj, uj+1, . . . , uj+p}
of consecutive integers in U (that is, uj−1 < uj−1; uj =uj+1−1; . . . ; uj+p−1 =
uj+p−1; uj+p < uj+p+1−1). Furthermore, consider the word w0ujw 0
uj+1. . . w
0
uj+p
of length at mostm, which is a factor of w0 = (ab)∞ and thus also a factor of aba because |aba| > 2m. The word aba, being a factor of w, appears infinitely often inw becausewis almost periodic. Hence not only we can define f :U →Nin such a way thatwf(u) =wu0 for any u ∈U and that blocks of consecutive integers in U
are mapped to blocks of consecutive integers, but we can also do it monotonically and so that f(u) > f(u0) + 1 whenever u > u0 + 1. This finishes the proof of the first implication.
(b)⇒ (c): Let X contain a class P(w, H), where w is a periodic word and H is a graph with loops allowed on the alphabet of w. If w is not cyclic (i.e., if some letters appear more than once within the period), the class P(w, H) can be transformed into a cyclic one by extending the alphabet, renaming multiple appearances of the same letter within the period to different letters of the extended alphabet and modifying the graph H accordingly. More formally, let p be the period ofw. Define a new word w0 = (123. . . p)∞ and a graphH0 with vertex set
{1,2, . . . , p} and an edge ij ∈E(H0) if and only if wiwj ∈E(H). Then any graph
Gw,H(u1, u2, . . . , um)∈ P(w, H) is isomorphic to the graphGw0,H0(u1, u2, . . . , um)∈
P(w0, H0). Hence X also containsP(w0, H0), where the wordw0 is cyclic.
Since any periodic word of periodpis also a periodic word of periodkpfor any k∈N, by the same transformation a periodic class of period p can be transformed into a cyclic class of periodkp.
(c)⇒(d): Follows directly from the definition of a cyclic class.
(d) ⇒ (a): Let X contain the `-factor Gw,H(1,2, . . . ,2`m). We prove that
X then contains the whole class P(w, H); its speed will then be above the Bell number by Proposition 5.3.1. For the sake of contradiction suppose that for somei, the graph Fi belongs to P(w, H). Then Fi ∼= Gw,H(u1, u2, . . . , uk) for some k ≤
m and u1 < u2 < · · · < uk. Note that the period of w is ` (by the definition
of a cyclic class). Put u01 = u1mod`. Furthermore, for each i ≥ 2 let u0i =
(ui mod`) + ci`, where each ci is chosen in such a way that 0 < u0i −u0i−1 ≤
`+ 2 for all i and u0i −u0i−1 = 1 if and only if ui −ui−1 = 1. By construction,
Fi ∼=Gw,H(u1, u2, . . . , uk) =∼Gw,H(u01, u02, . . . , u0k) and u
0
k < k(`+ 1) ≤m(`+ 1) ≤
2`m. HenceFi is isomorphic to an induced subgraph ofGw,H(1,2, . . . ,2`m)∈ X =
Free(F1, F2, . . . , Fn), a contradiction.
(c)⇒ (e): LetX contain the cyclic class P(w, H), where w is a cyclic word with periodp > `. Since p > `, the subgraph ofGw,H(1,2, . . . , pM) induced by the
bags corresponding to the first`letters of w is an (`, M)-strip.
(e)⇒ (a): Let `X, dX and cX be the constants given by Lemma 5.2.5 and putM = 2dX +cX + 2. We show that for any fixed positive integer `, the class X contains a strong (`X, dX)-graphGsuch that its sparsificationφ(G) has a component of order at least`. Then we can apply Theorem 5.2.2.
So let`be a positive integer. By assumption,X contains an (`, M)-stripSH,M.
By Lemma 5.2.5, after removing no more thancX vertices we are left with a strong (`X, dX)-graph S0 with a strong (`X, dX)-partition π. Let Va = {(a, j) ∈ V(S0) :
1 ≤ j ≤ M} be the letter bags of S0, 1 ≤ a ≤ `, and consider the prime parti- tionp(π) ={W1, W2, . . . , W`0}. If two vertices x, ybelong to different bags of p(π),
then according to Lemma 5.2.3 we have |N(x) N(y)| ≥ 5×2`Xd
X −2dX. But notice that if we have two vertices (a, j), (a, j0) of S0 in the same letter bag Va,
thenN(a, j) N(a, j0)⊆ {(a−1, j),(a−1, j0),(a+ 1, j),(a+ 1, j0)}, so its size is at most 4. Hence, we deduce thatVa⊆Wf(a) for some functionf.
Now notice that (Va, Vb) is dX-dense, that is, ab is and edge of H, if and only if (Wf(a), Wf(b)) is dX-dense. Indeed, if one of them was dX-dense and the other dX-sparse, then (Va, Vb) would be both dX-dense and dX-sparse, in which case |Va| ≤ 2dX + 1. But this is not true, as Va is obtained from a set of size
M = 2dX + 2 +cX by removing at mostcX vertices.
It follows that φ(S0) is constructed by swapping edges with non-edges be- tween Va and Vb such that ab ∈ E(H). Hence φ(S0) is a linear forest obtained
from the paths (1, j)−(2, j)− · · · −(`, j) for j ∈ {1,2, . . . , M} by removing at most cX vertices. As M > cX, at least one of the paths is left untouched. Therefore,
φ(S0) contains a component of size at least`.
Finally, we are ready to tackle the decidability of Problem 5.4.1. Algorithm 5.4.3.
Output: Yes, if the speed of X = Free(F) is above the Bell number; no otherwise.
(1) Using Theorem 5.4.1, decide whetherkX =∞. If it is, output yesand stop.
(2) Set m:= max{|F1|,|F2|, . . . ,|Fn|} and `:= 1.
(3) Loop:
(3a) For each graph (with loops allowed)H on{1,2, . . . , `} construct the(`, `)-
strip SH,`. Check if some Fi is an induced subgraph ofSH,`. If for each H
the stripSH,` contains some Fi, output no and stop.
(3b) For each graph (with loops allowed)H on{1,2, . . . , `}and for each wordw
consisting of`distinct letters from{1,2, . . . , `}check if the`-factorGw∞,H(1,
2, . . . ,2`m) contains some Fi as an induced subgraph. If one of these `-
factors contains noFi, output yes and stop.
(3c) Set`:=`+ 1and repeat.
It remains to prove the correctness of this algorithm. Theorem 5.4.4. Algorithm 5.4.3 correctly solves Problem 5.4.1.
Proof. We show that if the algorithm stops, it gives the correct answer, and fur-
thermore that it will stop on any input without entering an infinite loop. First, if it stops in step (1), the answer is correct by [Balogh et al., 2005], since any class with infinite distinguishing number has speed above the Bell number.
Assume that the algorithm stops in step (3a) and outputsno. This is because every (`, `)-strip contains some forbidden subgraphFi, hence no (`, `)-strip belongs
toX. By Theorem 5.4.2(e), the speed ofX is below the Bell number.
Next suppose that the algorithm stops in step (3b) and answers yes. Then
X contains the`-factorGw∞,H(1,2, . . . ,2`m), wherew∞ is a cyclic word. Hence by
Theorem 5.4.2(d) the speed of X is above the Bell number.
Finally, if kX = ∞ the algorithm stops in step (1). If kX < ∞ and the speed ofX is above the Bell number, then by Theorem 5.4.2(d) the algorithm will stop in step (3b). If, on the other hand, the speed of X is below the Bell number, then by Theorem 5.4.2(e) there exist positive integers `, M such that X contains no (`, M)-strip. Let N = max{`, M}. Obviously, X contains no (N, N)-strip, because any (N, N)-strip contains some (many) (`, M)-strips as induced subgraphs and X is hereditary. Therefore the algorithm will stop in step (3a) after finitely many steps.