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Cutsets of Boolean lattices
231
Discrete Mathematics 63 (1987) 231-240 North-Holland
CUTSETS OF BOOLEAN
LATTICES
Richard NOWAKOWSKI* Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4H8
Received 20 May 1986 Let a be an element of a finite ordered set P. A subset F of P is a cutset for a if every element of F is incomparable to a and if every maximal chain of P intersects F U {a}. The cardinalities of minimum sized cutsets for elements of finite boolean lattices are determined
1. Introduction
In graph theory, Menger’s Theorem is the basis of many results concerning notions of connectedness. There are many versions, in this paper the following will be useful. Let G = (V, E) be a directed graph and A, B disjoint subsets of V with no directed edge from A to B. A subset of V -A U B is called an (A, B) separating vertex-set if every directed path from A to B intersects F. A set of directed paths from A to B are called vertex disjoint if any two have at most one vertex of A and one vertex of B in common. Theorem 1. Let G = (V, E) be a directed graph and A, B G V. The minumum size of an (A, B) separating vertex-set equals the maximum number of vertex disjoint paths from A to B.
For a proof see Aigner [l, p. 3921, also see this book for any undefined graph theoretic concepts contained in this paper. In ordered sets, two concepts have been introduced which in the finite case can be related to questions of connectivity in the appropriate associated directed graph. In an ordered set (P, c), F is called a cutset if every maximal chain intersects F. If a E P, then F is called a cutset for a if F U {a} is a cutset and if no element of F is comparable to a. See Fig. l(a), (b) respectively, in each case the elements of F are indicated by dark circles. Note that in neither case does F have to be an antichain. Grillet [5], however, found a forbidden configuration characterization for certain ordered sets (the infinite chains satisfied some ‘regularity’ conditions) in which every maximal antichain is a cutset. Rival and Zaguia [6] characterized, again by means of a forbidden configuration, those ordered sets which can be decomposed into antichain cutsets. * Supported in part by NSERC Grant A4820. 0012-365X/87/$3.50 @ 1987, Elsevier Science Publishers B.V. (North-Holland)
232
R. Nowakowski
Y
x
I\yu/ RI a
z
{x,y, 2)
(4
x
@I
is a cutset.
{x, y} is a cutset for a.
Fig. 1.
The idea of cutsets for elements was introduced by Bell and Ginsburg [2] while investigating the space of maximal chains in the product topology. As with the first notion of cutsets, the main thrust in the investigation of cutsets for elements has been structural, mainly relating them to antichains (see Sauer and Woodrow [7] and Ginsburg, Rival and Sands [4]). In this paper however a quantative approach is taken based on Theorem 1. Let a, b E P, (P, C) an ordered set, a covers b, written a > b, if a > b and a > c 3 b implies that b = c. The directed diagram of (P, G) is a directed graph D(P) = (P, E) w h ere (b, a) E E if a covers b. Note that any maximal chain between two elements of P corresponds to a directed path joining them in D(P). Let Pb = P U (0, l},where Pb inherits the order of P and in addition 0 sx < 1 for all x E P. If p has one or both bounds then (0, l} are not necessarily disjoint from P. Let P’ be the ordered set obtained when 0 and 1 are disjoint from P (see Fig. 2). Since any maximal chain between two elements of P corresponds to a directed path joining them in D(P), the proof of the next result is immediate. Theorem 2. Let (P, C) be a finite ordered set and F c P. F is a cutset for P if and only if F is a (0, 1)separating vertex-set of D(P”). F is a cutset for a if and only if F U {a} is a ([0, a), (a, 11)separating vertex-set in D(Pb). In the sequel all ordered sets will be assumed to be finite and bounded. The latter is no restriction because of the preceeding result. Let x E P and {c} be the collection of cutsets for x. Let C(u, P) = maxi {I&]}. If P is finite then C(u, P) can be determined efficiently using Theorem 2 and the network flow algorithms associated with Menger type results. For arbitrary ordered sets this may be the only approach possible. However, if P is ‘regular’, then it may be possible to give an explicit formula for C(u, P), in terms of other invariants of P, and also to describe such cutsets. In Pb, let U(u) = {y E P 1y > a, yu} and I(u)={y~PlyCu;y>z for some ~
Cutsetsof Boolean lattices
233
D(Pb)
D(P’) Fig. 2.
C(a, P), how good these bounds are depends on P. However, it would seem that such cutsets would be close to optimal if P was unimodular ([l, p. 4191). No general results are known except for 2-dimensional distributive lattices (Rival, Zaguia [6]) in which min{(U(a)], IL(a)]} = C(a, P). Let 2” denote the Boolean lattice on iz elements. Let a, b E 2”, if h(a) = h(b) then there is an automorphism of 2” which maps a to b. Therefore the size and structure of cutsets for a particular element in a Boolean lattice depends on the heights of the element. Consequently let C(i, n) denote the size of the smallest cutset associated with any (and hence all) elements of height i in 2”. Theorem n=4,
3. Let a E 2”, h(u) = i, then C(i, n) = min{]L(u)], ]~(a)]} i=3.
except if
i=2orifn=6,
If a, b E 2” and b < a, then b has n - h(u) upper covers which are not less than a. Since [0, a] = 2 ‘@) it follows that IL(u)1 = (n - h(u)(2h(“) - 1). Dually, it can be shown that )~(a)[ = h(u)(2”-h(“) - 1). Theorem 3 can be rephrased as Corollary 4. C(i, n) = min{(n - i)(2’ - l), i(2”-’ - 1)) provided ornZ6, if3. 2. Proof of Theorem
that n f 4, i f 2
3
The proof proceeds by induction, however the induction step involves two cases. Determining C(i, n) for i = $z and i # +n have to be considered separately, The case i # $z is simpler and is discussed first.
R. Nowakowski
234
Throughout the sequel, the atoms of 2” will be denoted a,, u2, . , . , a, and the element of height i will be taken as a = //jzl Uj. Even though the proof uses the directed graph D(2”) it will be useful to retain the notion of the height of an element as measured in 2”. In fact if P is a directed path P(i) will be used to denote the element of height i in the path and P(i, i) the segment of P from the element of height i to that of height i. (In this section all paths are directed paths and so the adjective will be dropped.) Also, if a is an atom and P = {b,, b1, . . . > bk} is a path then P v a will denote the path {bO v a, b1 v a 3 . . . 7 bk v a}. Lemma
5. For n 5 4, i 6 $z if C(i, n) = (n - 1)(2’ - l), then C(i, n + 1) = (n + 1 - i)(2’ - 1). Similarly if i 5 $2 and C(i, n) = i(2n-’ - l), then C(i + 1, n + 1) =
(i + 1)(2”_’ - 1). Proof.
Let a,, a2, . . . , a,,, be the atoms of 2”+i and let u = Viz1 Uj. Let A be the sublattice generated by {ul, a2, . . . , a,} so that 2”+’ =A U {A v a,+I}. In A, h(u) = i. Suppose that C(i, n) = (n - i)(2’ - 1) then by Menger’s Theorem there is a family 3 of (n - i)(2’ - 1) disjoint paths in A from [0, a) to (a, 11. For each b E [0, a) choose a path, Pb of 9, that starts at b. If Pb = {b. = bR}, then let PL = {b,, b. v a,,,, bI v u,+~, . . . , bR v a,,,} and 6, b,, bz, . . . , let 9 = {Pk 1b E [0, a)}. Now in 2”+l, the paths of 4 U 2 are disjoint and connect elements below a to elements above a. Moreover the upper covers of [0, a) in A and of [0, a) v ancl are, respectively, each contained in a unique path of 3, 8. Hence there are IL(u)\ = (n + 1 - i)(2’ - 1) disjoint paths between [0, a) and (a, l] in 2”+l and the first result now follows from Menger’s Theorem. The second half follows from the dual version of the preceeding argument. 0 If n is even then Lemma 5 gives C(i, n + 1) for all i. If n is odd however then C(i, n -t 1) is given for all i except i = i(n + l), this case is covered next. Lemma Proof.
6. In the lattice 22m, m #2,
3, C(m, 2m) = m(2” - 1)
the result is trivial for m = 1 and it is easy to show that C(2, 4) = 5 and C(3, 6) = 19. Table 1 lists 4(2” - 1) = 60 disjoint paths joining L(b) and U(b) where b = Vj’zl aj. The table uses characteristic vectors to represent the elements where the first four places correspond to atoms below 6; the elements of a path are read from left to right and a dash means the four coordinates have not changed. These paths can be extended to paths from [0, b) to (b, l] by the addition of the obvious two elements, In general, the path from [0, b) to (b, l] corresponding to a path P from L(b) to U(b), will be called the extension of P and is denoted by p. Assume that for some m > 4, C(m, 2m) = m(2” - 1). Let b = V~Z~ aj. Assume also that there is a family $ of disjoint paths from L(b) to U(b), ]9]= m(2” - l),
Cutsets of Boolean lattices
235
Table 1 1110,1000 1110,0100P[3, l] 1110,0010 1110,0001 1101,1000 P[3,3] 1101,0100 1101,0010 1101,0001 1011,1000 1011,0100 1011,0010 1011,oool P[3, 41 0111,1000 0111,0100 P[3, 21 0111,0010 0111,0001
1100,1000 -) 1100,0100 -, 1100,0010 -, 1100,COO1-, 1010,1000 -, 1010,0100 -) 1010,0010 -, 1010,0001 -, 1001,1000 -, 1001,0100 -, 1001,0010 -, lOOl,OOOl-, 0011, 1000 -, 0011,0100 -, 0011,0010 -, 0011,0001 -) 0101,1000 -, 0101,0100 -, 0101,0010 -, 0101,0001 -, 0101,1000 -, 0110,0100 -, 0110,0010 -, 0110,0001 -,
looo, 1000 -, 11000 1000,0100 -, 0110 1000,0010 -, 0011 1000,0001 -, 1001 0100,1000 -) 1100 0100,0100 -, 0110 0100,0010 -, 1010 0100,0001 -, 1001 0010,1000 -, 1100 0010,0100 -) 0110 0010,0010 -, 0011 0010,0001 -, 0101 0001, 1000 -, 1001 0001,0100 -) 1100 0001,0010 -, 0110 0001,0001 -, 0011
1001,1100 -, 0111 -) 1011 -, 1101 -, 1110 0110,0110 -, 1011 -, 1101 -, 1110 -, 0111
0000,loo0 -, 1100 0110 0000,0101 -, 0011 0000,Oool -, 1001
1110 0111 1011 1101
0000, 0100 -,
0011, 0011
-, 1101 -, 1011 1001, 1100
-, -, -, -,
where 4 contains such that each pij distinct element of 1, 2, . . . , m such Pi_1,4tij(m + 1) for
-, 1110 -, 0111
1100 0110 0011 1001 1010 1100 0110 0101 1001 0101 1010 0011 1100 0110 1010 1001 1010 1100 0110 0101 1001 1100 1010 0011
1101,1100 1110,0110 1110,0011 1101,1001 P[2,3] 1110,1010 1011,1100 1011,0110 1011,0101 1011,1001P[2,4] 1101,0101 1101,1010 1101,0011 1011,1100 0111,0110P[2,2] 1011,1010 0111,1001 1101,1010 0111,1100 1101,0110 0111,0101 1110,1001 1110,1100P[2, l] 0111,1010 0111,0011
-) 1110 1100,0111 1100,1011 1100,1101 0110,1110 -, 0111 0110,1011 0110,1101 0011,1110 0011,0111 -, 1011 0011,1101 1001,1011 1001,1101 1001,1110 1001,0111
1101,1110 1110,0111 1110,1011 1110,1101P[l, 1110,1110P[l, 0111,0111 0111,1011 0111,1101 0111,1110 1011,0111P[l, 1011,1011 1011,1101 1101,1011P[l, 1101,1101 1011,1110 1101,0111
1000,1110 -, 1111 llOO,0100,0111 -, 1111 0110,0010,1011 -, 1111 0011,Oool, 1101 -, 1111 lOOl,-
31 31
21 41
lllO,0111,1011,llOl,-
Q[3] = P[l] Q[l] = P[2] Q[2] = P[4] Q[4]=P[3]
a subfamily S={f$li=O, 1,. . . ,m-l,j=1,2,. . . ,m} starts at a distinct element of height i and terminates at a height 2m - i. Moreover, there exists permutations nip oi of that ni(l) = 1 = q(l) and Pi_l,n,uj(m - 1) -C$(m) -C i = 2, 3, . . . , m - 1, j = 1, 2, . . . , m.
236
R. Nowakowski
In addition, assume that there are two families 9 = {Qi} and LB= {&} j = 1,2, ...,m of disjoint paths from 0 to 1.These paths intersect exactly one path of 9, specifically P&y moreover RI = Q, = &. The paths also have the following relationships: (i) Rj(m - 1) < Pr,j(m) < Qj(m + 1); (ii) R,(m - 1) = Qr(m - 1) < V {ai 1ai d x, ai at atom of Zzm} < R,(m - 1) =
Qdm + 1). For m = 4, there paths are given in Table 1. The paths in 9, 9 and 5%are important. The elements of these paths can be rearranged to form new ‘twisted’ paths which do not intersect any other paths of 9. Figure 3(a) gives an idea of the roles these three families play. An element x of a path in the lattice occupies position (p, q) if h(x) = p + q and h(x A b) = q. If each of the paths had elements that could be represented (in general) as (1, q), (2, q), . . . 7 (m - 4, q), (m - 4, q + 11, Cm - 4,q + 3,. . . , Cm - 9, m - I), then the paths would be represented by the dark lines of the figure and the lemma would be proved. However there is only one element that can occupy (4,O) and the paths starting at (1,0) must find another route. The family 9 follow the dark lines, the families .% and 9 follow the wavy lines and are so chosen that at (3,l) the paths can be reformed to follow those indicated in Fig. 3(b). One problem is to recover these families at the end of the induction step.
(a)
(b) Fig. 3.
Cutsets of Boolean lattices
237
Let {ai 1i = 1,2, . . . , 2m + 2) be the atoms of 22m+2 and let b1 = Vz<’ ai. The rest of the proof is concerned with constructing (m + 1)(2m+1 - 1) paths from W,) to VW. Let z = V {ai 1i # m + 1,2m + 2) and A = [0, z]. The lattice 22m+2 can be decomposed into four subsets A0 =A X {0}, Al =A X {u,+~}, A2=A X {~2~+~} and A3 =A X {a,,, v u~~+~}, each isomorphic to 22m. Let b = bo = V:l Uip b3 = b, v 6, and y. = vF?Z’z:, a,, y1= yo v %+1, Y2 = Yo v a2mt2, b2 = bo v a2m+2, y3 = y, v y,. Note that bi, yi E Ai for i = 0, 1, 2, 3. The disjoint paths needed to show that C(m + 1, 2m + 2) = (m + 1)(2m+1 - 1) will be constructed in parts by using the family $ of disjoint paths in 22”. Since IU(b,)l = IL(b,)l = (m + 1)(2m+1 - 1) it will suffice to construct disjoint paths such that each path contains precisely one element of U(b,). In 22m+2, b. < bl < b3 and b,< b2< b3 and in Ai, h(bi) = m, i = 0, 1,2,3. Therefore let 3 be the family of disjoint paths for b. in A0 given by induction and let {Qj} and {Pi} be the other assumed families. Let 2 be a family of 22m- 2 paths of $ which includes {P,j ( i = 1, 2, . . . , m - 1, j = 1,2, . . . , m} such that every element of (0, b,) and (b,, z) is contained in precisely one extension of the 4’s.
The majority of the new paths are described in (1) and (2). (1) The paths of {P v u,+~ 1P E 9 - {Poj 1j = 1,2, . . . , m}}. Let 9’=9-($U{Po,jIj=1,2,. . . , m}). If p is a path of $‘, let p, p’, p” be the first, penultimate and last vertices, respectively, of P. (2) The {P” v
a2m+2
paths v
%+I~
of
{PVG),
I p E 9’1.
M.P’)) U {P’ v %+2)
U {P’ v a~,,,+2 v a,+d
U
These paths are disjoint and each contains exactly one vertex of L(b,). The vertices of L(b,) which are unaccounted for are [0, b,) v u2m+2, the upper covers of b. in A0 and the vertices of U(b,) contained in the paths of 9 U {Po,j 1j = 1, 2, . . . , m}. The other paths are constructed as follows. In A0 take the paths segments (3) 8; - {P,j 1i = 1, 2, . . . , m - 1, j = 1, 2, . . . , m}; (4) e,j(i, m) U Pi-l,o,o-)(m + 1, 2m - i + l), i = 2, 3, . . . , m - 1, j = 1, 2, . . . , (5) :;(I,
m) U Q,(m + l), j = 2, 3, . . . , m;
(6) S:dL m) U Q,(m + 1, 2m);’ (7) P,,j(O, m), i=L 2,. . . , m; (8) {bo, bo v ai, bl v ai}, i = m + 2, m + 3, . . . , 2m + 1. Since Q,(m + 1) > P,,,(m) the set constructed in (6) is a path. Also all the paths are disjoint. The upper covers of [0, b,) are each contained in precisely one path of those in (2), (3), . . . , (8). The upper covers of b. in A0 are upper covers of [0, b,) in 22m+2, these are disposed of in (8). In Al, take the path segments (9) Rj(O, m - l), j = 2, 3, . . . , m;
238
R. Nowakowski
(10) RI@, m - 1)u {YI>u QI<~ + 1, 2~); (11) (Continuation of (5)) Qj(m + 1, 2m), j = 2, 3, . . . , m; (12) (Continuation of part of (7)) P,,,(m). Since {Rj} and {Qj} only intersect {Po,j}, the paths contained in (9)-(12) are mutually disjoint, moreover since {P,,j(l)} = {Rj(l)} every element of L(b,) flAl is contained in precisely one path of (l), (9), (lo), (ll), (12). In A2 take, (13) 9; (14) PO,,; (15) (Continuation (16) {b,
of (7)) PO,j(WZ, 2m + l), j = 2, 3, . . f, m;
bz v em+d;
By the choice of 9 and {Po,j}, all U(b,) n & = bm+z> &I is contained and (16). In A-,, essentially the construction reversed. Specifically, take (17) 8;-{S,jli=l,2,...,m,j=l,2,...,m}; (18) Pi,,i+,G,(i, m - 1) U Pi+l,j(m,
the paths are disjoint and every element of in precisely one path of (2), (13), (14), (15) in Al using the E,j’s, the Q’s and the R’s, is
i = 1, 2, . . . , m - 2,
2m - i - l),
j = 1,
(19) ~~,;;vmiVi,ai),I~i~m-tl,j=l.2,...,m}; (20) (Continuation of (9)) Rj(m - 1) U Pi,j(m, 2m - l), j = 2, 3, . . . , m; (21) R1(O, m - 1) U Pl,l(m, 2m - 1); (22) (Continuation of (15)) Po,j(m + 1, 2m), j = 2, 3, . . . , m; (23) (Continuation of (12)) P,,,(m, 2m). Again, all the paths are disjoint and each element of U(b,) fTA, = [a,,, v u~,,,+~, b3) is contained in precisely one path of (17), (18), (19) and (21). The preceeding construction has produced IL(b,)l may disjoint paths from L(b,) to U(b,) and consequently the extensions of these paths form [L(bJl many disjoint paths from [0, b) to (b, 11.It remains to identify in 22m-c2the families of paths S’, 3’ and B’ corresponding to 2, L?iZand 9 and give the appropriate permutations 0; and nl. The family of paths .Y’={Pk(i=O,l,,.., m,j=1,2 ,..., m+l} with the permutations are pl+l,j+l
=
Ej
V %+l,
4+l(i
+
1)
=
Qi)
fort=~,0,~=1,2 PI+1,* =Pi1vuti+2,
tl(l)=l
Ph, = {b,, bZ, b3}, t;(l)
fort=Jd,
,..., a,i=l,2,.
m-l,j=1,2
,...,
m;
,...,
m;
. . ,m-1;
= 1 for t = zr, a;
P;j= Plj(l, m) U Qj(m + 1) U (Qj(m + 1, 2m) v a,+,),z;(j)= j
forr=n,a,j=2,3
Cutsets of Boolean lattices
P& =
poi(O,
m + 1) V U2m+2)
m) U (POj(m,
j=2,3,.
U(P,i(m+1,2m)VU,+~VU~,+~),
P&,+1 = P,,(O, m) U (P&(m) v u.,+J %
239
U (P,,(m,
. . ,m;
2m) v u,+l v uh+2);
= (&I v uzm+d*
of al and xf are easy to verify for i 3 2 since Pk(m + 1) = Pi_l,j_l(WZ) V U,+l (if j > 1). If i Z m and j = 1 then Pil(m + 1) = Pi_l,l(m) v ati+* which is covered by Pi_l,l(m + 2) and covers Pf_l,l(m). If i = m, then - 1) v u~,,,+~ and Pkl(m + 1) = bI = b. v a&+2. However Pk_l, l(m) = P,_l,l(m P,_l,l(m - 1) is a lower cover of bO, i.e., PLl(m + 1) > Pk_l,l(m). In a similar but dual manner it can be shown that Pk_,,,(m + 2) > PLl(m + 1). The family 9’ is defined as follows. The properties
Qi’ = Qj(‘J, ml U (Qj(m, m + 1) v U
~2~+2)
(Qj(m + 1,2m) v a,+~ v a~~+2),
j = 2,3, . . . , m;
Qi = QI@, 2m) v ~2~+2; Q ;+I = QI<% m - 1) U U
(~~1 U {y2) U {y3)
The family 9’ is given by R; = Q;,
forj=2,3,...,m+l;
R; = (R,(O, 2m) v azrn+2) In particular, note that R, = Ql = PO, so that RI = Q; = P&. Since P,,(m - 1) < yO< P,,(m + l), it follows that P&(m) = P,,(m - 1) v uzmc2-Cy2 -KPol(m + 1) v U 2m+2=P&(m+2). Moreover, for j=2,3,. . . , m, the following covering relations hold: Qi(m + 2) = Qj(m + 1)
v uzrn+2)
Qj(m + 1)
= PQ(m + 1) > Q,(m) = R;(m);
Qi pII = Pil(m + 1) > R,(m - 1) v Qi,,+l
y3 ) y1 = PL+dm
v em+2 u2m+2= R;(m);
+ 1) > y. =
QL+l(m).
It is easy to verify that the paths from 9’, respectively 9?.‘, each intersect a unique path of Phj. The required families 2’, 9’ and 9’ have been found, together with the appropriate covering relations. Consequently Lemma 6 has been proved and hence so has Theorem 3. 0
240
R. Nowakowski
References [l] M. Aigner, Combinatorial Theory (Springer, New York, 1979). [2] M. Bell and J. Ginsburg, Compact spaces and spaces of maximal complete subgraphs, Trans. Amer. Math. Sot. 283 (1984) 329-338. [3] J. Ginsburg, Compactness and subsets of ordered sets that meet all maximal chains, Order 1 (1984) 147-1.57. [4] J. Ginsburg, I. Rival and B. Sands, Antichains and finite sets that meet all maximal chains, to appear. [5] P.A. Grillet, Maximal chains and antichains, Fund. Math. 65 (1969) 157-167. [6] I. Rival and N. Zaguia, Crosscuts and cutsets, Order 1 (1984) 235-247. [7] N. Sauer and R. Woodrow, Finate cutsets and finite antichains, Order 1 (1984) 35-46.
Discrete Mathematics 63 (1987) 231-240 North-Holland
CUTSETS OF BOOLEAN
LATTICES
Richard NOWAKOWSKI* Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4H8
Received 20 May 1986 Let a be an element of a finite ordered set P. A subset F of P is a cutset for a if every element of F is incomparable to a and if every maximal chain of P intersects F U {a}. The cardinalities of minimum sized cutsets for elements of finite boolean lattices are determined
1. Introduction
In graph theory, Menger’s Theorem is the basis of many results concerning notions of connectedness. There are many versions, in this paper the following will be useful. Let G = (V, E) be a directed graph and A, B disjoint subsets of V with no directed edge from A to B. A subset of V -A U B is called an (A, B) separating vertex-set if every directed path from A to B intersects F. A set of directed paths from A to B are called vertex disjoint if any two have at most one vertex of A and one vertex of B in common. Theorem 1. Let G = (V, E) be a directed graph and A, B G V. The minumum size of an (A, B) separating vertex-set equals the maximum number of vertex disjoint paths from A to B.
For a proof see Aigner [l, p. 3921, also see this book for any undefined graph theoretic concepts contained in this paper. In ordered sets, two concepts have been introduced which in the finite case can be related to questions of connectivity in the appropriate associated directed graph. In an ordered set (P, c), F is called a cutset if every maximal chain intersects F. If a E P, then F is called a cutset for a if F U {a} is a cutset and if no element of F is comparable to a. See Fig. l(a), (b) respectively, in each case the elements of F are indicated by dark circles. Note that in neither case does F have to be an antichain. Grillet [5], however, found a forbidden configuration characterization for certain ordered sets (the infinite chains satisfied some ‘regularity’ conditions) in which every maximal antichain is a cutset. Rival and Zaguia [6] characterized, again by means of a forbidden configuration, those ordered sets which can be decomposed into antichain cutsets. * Supported in part by NSERC Grant A4820. 0012-365X/87/$3.50 @ 1987, Elsevier Science Publishers B.V. (North-Holland)
232
R. Nowakowski
Y
x
I\yu/ RI a
z
{x,y, 2)
(4
x
@I
is a cutset.
{x, y} is a cutset for a.
Fig. 1.
The idea of cutsets for elements was introduced by Bell and Ginsburg [2] while investigating the space of maximal chains in the product topology. As with the first notion of cutsets, the main thrust in the investigation of cutsets for elements has been structural, mainly relating them to antichains (see Sauer and Woodrow [7] and Ginsburg, Rival and Sands [4]). In this paper however a quantative approach is taken based on Theorem 1. Let a, b E P, (P, C) an ordered set, a covers b, written a > b, if a > b and a > c 3 b implies that b = c. The directed diagram of (P, G) is a directed graph D(P) = (P, E) w h ere (b, a) E E if a covers b. Note that any maximal chain between two elements of P corresponds to a directed path joining them in D(P). Let Pb = P U (0, l},where Pb inherits the order of P and in addition 0 sx < 1 for all x E P. If p has one or both bounds then (0, l} are not necessarily disjoint from P. Let P’ be the ordered set obtained when 0 and 1 are disjoint from P (see Fig. 2). Since any maximal chain between two elements of P corresponds to a directed path joining them in D(P), the proof of the next result is immediate. Theorem 2. Let (P, C) be a finite ordered set and F c P. F is a cutset for P if and only if F is a (0, 1)separating vertex-set of D(P”). F is a cutset for a if and only if F U {a} is a ([0, a), (a, 11)separating vertex-set in D(Pb). In the sequel all ordered sets will be assumed to be finite and bounded. The latter is no restriction because of the preceeding result. Let x E P and {c} be the collection of cutsets for x. Let C(u, P) = maxi {I&]}. If P is finite then C(u, P) can be determined efficiently using Theorem 2 and the network flow algorithms associated with Menger type results. For arbitrary ordered sets this may be the only approach possible. However, if P is ‘regular’, then it may be possible to give an explicit formula for C(u, P), in terms of other invariants of P, and also to describe such cutsets. In Pb, let U(u) = {y E P 1y > a, y
Cutsetsof Boolean lattices
233
D(Pb)
D(P’) Fig. 2.
C(a, P), how good these bounds are depends on P. However, it would seem that such cutsets would be close to optimal if P was unimodular ([l, p. 4191). No general results are known except for 2-dimensional distributive lattices (Rival, Zaguia [6]) in which min{(U(a)], IL(a)]} = C(a, P). Let 2” denote the Boolean lattice on iz elements. Let a, b E 2”, if h(a) = h(b) then there is an automorphism of 2” which maps a to b. Therefore the size and structure of cutsets for a particular element in a Boolean lattice depends on the heights of the element. Consequently let C(i, n) denote the size of the smallest cutset associated with any (and hence all) elements of height i in 2”. Theorem n=4,
3. Let a E 2”, h(u) = i, then C(i, n) = min{]L(u)], ]~(a)]} i=3.
except if
i=2orifn=6,
If a, b E 2” and b < a, then b has n - h(u) upper covers which are not less than a. Since [0, a] = 2 ‘@) it follows that IL(u)1 = (n - h(u)(2h(“) - 1). Dually, it can be shown that )~(a)[ = h(u)(2”-h(“) - 1). Theorem 3 can be rephrased as Corollary 4. C(i, n) = min{(n - i)(2’ - l), i(2”-’ - 1)) provided ornZ6, if3. 2. Proof of Theorem
that n f 4, i f 2
3
The proof proceeds by induction, however the induction step involves two cases. Determining C(i, n) for i = $z and i # +n have to be considered separately, The case i # $z is simpler and is discussed first.
R. Nowakowski
234
Throughout the sequel, the atoms of 2” will be denoted a,, u2, . , . , a, and the element of height i will be taken as a = //jzl Uj. Even though the proof uses the directed graph D(2”) it will be useful to retain the notion of the height of an element as measured in 2”. In fact if P is a directed path P(i) will be used to denote the element of height i in the path and P(i, i) the segment of P from the element of height i to that of height i. (In this section all paths are directed paths and so the adjective will be dropped.) Also, if a is an atom and P = {b,, b1, . . . > bk} is a path then P v a will denote the path {bO v a, b1 v a 3 . . . 7 bk v a}. Lemma
5. For n 5 4, i 6 $z if C(i, n) = (n - 1)(2’ - l), then C(i, n + 1) = (n + 1 - i)(2’ - 1). Similarly if i 5 $2 and C(i, n) = i(2n-’ - l), then C(i + 1, n + 1) =
(i + 1)(2”_’ - 1). Proof.
Let a,, a2, . . . , a,,, be the atoms of 2”+i and let u = Viz1 Uj. Let A be the sublattice generated by {ul, a2, . . . , a,} so that 2”+’ =A U {A v a,+I}. In A, h(u) = i. Suppose that C(i, n) = (n - i)(2’ - 1) then by Menger’s Theorem there is a family 3 of (n - i)(2’ - 1) disjoint paths in A from [0, a) to (a, 11. For each b E [0, a) choose a path, Pb of 9, that starts at b. If Pb = {b. = bR}, then let PL = {b,, b. v a,,,, bI v u,+~, . . . , bR v a,,,} and 6, b,, bz, . . . , let 9 = {Pk 1b E [0, a)}. Now in 2”+l, the paths of 4 U 2 are disjoint and connect elements below a to elements above a. Moreover the upper covers of [0, a) in A and of [0, a) v ancl are, respectively, each contained in a unique path of 3, 8. Hence there are IL(u)\ = (n + 1 - i)(2’ - 1) disjoint paths between [0, a) and (a, l] in 2”+l and the first result now follows from Menger’s Theorem. The second half follows from the dual version of the preceeding argument. 0 If n is even then Lemma 5 gives C(i, n + 1) for all i. If n is odd however then C(i, n -t 1) is given for all i except i = i(n + l), this case is covered next. Lemma Proof.
6. In the lattice 22m, m #2,
3, C(m, 2m) = m(2” - 1)
the result is trivial for m = 1 and it is easy to show that C(2, 4) = 5 and C(3, 6) = 19. Table 1 lists 4(2” - 1) = 60 disjoint paths joining L(b) and U(b) where b = Vj’zl aj. The table uses characteristic vectors to represent the elements where the first four places correspond to atoms below 6; the elements of a path are read from left to right and a dash means the four coordinates have not changed. These paths can be extended to paths from [0, b) to (b, l] by the addition of the obvious two elements, In general, the path from [0, b) to (b, l] corresponding to a path P from L(b) to U(b), will be called the extension of P and is denoted by p. Assume that for some m > 4, C(m, 2m) = m(2” - 1). Let b = V~Z~ aj. Assume also that there is a family $ of disjoint paths from L(b) to U(b), ]9]= m(2” - l),
Cutsets of Boolean lattices
235
Table 1 1110,1000 1110,0100P[3, l] 1110,0010 1110,0001 1101,1000 P[3,3] 1101,0100 1101,0010 1101,0001 1011,1000 1011,0100 1011,0010 1011,oool P[3, 41 0111,1000 0111,0100 P[3, 21 0111,0010 0111,0001
1100,1000 -) 1100,0100 -, 1100,0010 -, 1100,COO1-, 1010,1000 -, 1010,0100 -) 1010,0010 -, 1010,0001 -, 1001,1000 -, 1001,0100 -, 1001,0010 -, lOOl,OOOl-, 0011, 1000 -, 0011,0100 -, 0011,0010 -, 0011,0001 -) 0101,1000 -, 0101,0100 -, 0101,0010 -, 0101,0001 -, 0101,1000 -, 0110,0100 -, 0110,0010 -, 0110,0001 -,
looo, 1000 -, 11000 1000,0100 -, 0110 1000,0010 -, 0011 1000,0001 -, 1001 0100,1000 -) 1100 0100,0100 -, 0110 0100,0010 -, 1010 0100,0001 -, 1001 0010,1000 -, 1100 0010,0100 -) 0110 0010,0010 -, 0011 0010,0001 -, 0101 0001, 1000 -, 1001 0001,0100 -) 1100 0001,0010 -, 0110 0001,0001 -, 0011
1001,1100 -, 0111 -) 1011 -, 1101 -, 1110 0110,0110 -, 1011 -, 1101 -, 1110 -, 0111
0000,loo0 -, 1100 0110 0000,0101 -, 0011 0000,Oool -, 1001
1110 0111 1011 1101
0000, 0100 -,
0011, 0011
-, 1101 -, 1011 1001, 1100
-, -, -, -,
where 4 contains such that each pij distinct element of 1, 2, . . . , m such Pi_1,4tij(m + 1) for
-, 1110 -, 0111
1100 0110 0011 1001 1010 1100 0110 0101 1001 0101 1010 0011 1100 0110 1010 1001 1010 1100 0110 0101 1001 1100 1010 0011
1101,1100 1110,0110 1110,0011 1101,1001 P[2,3] 1110,1010 1011,1100 1011,0110 1011,0101 1011,1001P[2,4] 1101,0101 1101,1010 1101,0011 1011,1100 0111,0110P[2,2] 1011,1010 0111,1001 1101,1010 0111,1100 1101,0110 0111,0101 1110,1001 1110,1100P[2, l] 0111,1010 0111,0011
-) 1110 1100,0111 1100,1011 1100,1101 0110,1110 -, 0111 0110,1011 0110,1101 0011,1110 0011,0111 -, 1011 0011,1101 1001,1011 1001,1101 1001,1110 1001,0111
1101,1110 1110,0111 1110,1011 1110,1101P[l, 1110,1110P[l, 0111,0111 0111,1011 0111,1101 0111,1110 1011,0111P[l, 1011,1011 1011,1101 1101,1011P[l, 1101,1101 1011,1110 1101,0111
1000,1110 -, 1111 llOO,0100,0111 -, 1111 0110,0010,1011 -, 1111 0011,Oool, 1101 -, 1111 lOOl,-
31 31
21 41
lllO,0111,1011,llOl,-
Q[3] = P[l] Q[l] = P[2] Q[2] = P[4] Q[4]=P[3]
a subfamily S={f$li=O, 1,. . . ,m-l,j=1,2,. . . ,m} starts at a distinct element of height i and terminates at a height 2m - i. Moreover, there exists permutations nip oi of that ni(l) = 1 = q(l) and Pi_l,n,uj(m - 1) -C$(m) -C i = 2, 3, . . . , m - 1, j = 1, 2, . . . , m.
236
R. Nowakowski
In addition, assume that there are two families 9 = {Qi} and LB= {&} j = 1,2, ...,m of disjoint paths from 0 to 1.These paths intersect exactly one path of 9, specifically P&y moreover RI = Q, = &. The paths also have the following relationships: (i) Rj(m - 1) < Pr,j(m) < Qj(m + 1); (ii) R,(m - 1) = Qr(m - 1) < V {ai 1ai d x, ai at atom of Zzm} < R,(m - 1) =
Qdm + 1). For m = 4, there paths are given in Table 1. The paths in 9, 9 and 5%are important. The elements of these paths can be rearranged to form new ‘twisted’ paths which do not intersect any other paths of 9. Figure 3(a) gives an idea of the roles these three families play. An element x of a path in the lattice occupies position (p, q) if h(x) = p + q and h(x A b) = q. If each of the paths had elements that could be represented (in general) as (1, q), (2, q), . . . 7 (m - 4, q), (m - 4, q + 11, Cm - 4,q + 3,. . . , Cm - 9, m - I), then the paths would be represented by the dark lines of the figure and the lemma would be proved. However there is only one element that can occupy (4,O) and the paths starting at (1,0) must find another route. The family 9 follow the dark lines, the families .% and 9 follow the wavy lines and are so chosen that at (3,l) the paths can be reformed to follow those indicated in Fig. 3(b). One problem is to recover these families at the end of the induction step.
(a)
(b) Fig. 3.
Cutsets of Boolean lattices
237
Let {ai 1i = 1,2, . . . , 2m + 2) be the atoms of 22m+2 and let b1 = Vz<’ ai. The rest of the proof is concerned with constructing (m + 1)(2m+1 - 1) paths from W,) to VW. Let z = V {ai 1i # m + 1,2m + 2) and A = [0, z]. The lattice 22m+2 can be decomposed into four subsets A0 =A X {0}, Al =A X {u,+~}, A2=A X {~2~+~} and A3 =A X {a,,, v u~~+~}, each isomorphic to 22m. Let b = bo = V:l Uip b3 = b, v 6, and y. = vF?Z’z:, a,, y1= yo v %+1, Y2 = Yo v a2mt2, b2 = bo v a2m+2, y3 = y, v y,. Note that bi, yi E Ai for i = 0, 1, 2, 3. The disjoint paths needed to show that C(m + 1, 2m + 2) = (m + 1)(2m+1 - 1) will be constructed in parts by using the family $ of disjoint paths in 22”. Since IU(b,)l = IL(b,)l = (m + 1)(2m+1 - 1) it will suffice to construct disjoint paths such that each path contains precisely one element of U(b,). In 22m+2, b. < bl < b3 and b,< b2< b3 and in Ai, h(bi) = m, i = 0, 1,2,3. Therefore let 3 be the family of disjoint paths for b. in A0 given by induction and let {Qj} and {Pi} be the other assumed families. Let 2 be a family of 22m- 2 paths of $ which includes {P,j ( i = 1, 2, . . . , m - 1, j = 1,2, . . . , m} such that every element of (0, b,) and (b,, z) is contained in precisely one extension of the 4’s.
The majority of the new paths are described in (1) and (2). (1) The paths of {P v u,+~ 1P E 9 - {Poj 1j = 1,2, . . . , m}}. Let 9’=9-($U{Po,jIj=1,2,. . . , m}). If p is a path of $‘, let p, p’, p” be the first, penultimate and last vertices, respectively, of P. (2) The {P” v
a2m+2
paths v
%+I~
of
{PVG),
I p E 9’1.
M.P’)) U {P’ v %+2)
U {P’ v a~,,,+2 v a,+d
U
These paths are disjoint and each contains exactly one vertex of L(b,). The vertices of L(b,) which are unaccounted for are [0, b,) v u2m+2, the upper covers of b. in A0 and the vertices of U(b,) contained in the paths of 9 U {Po,j 1j = 1, 2, . . . , m}. The other paths are constructed as follows. In A0 take the paths segments (3) 8; - {P,j 1i = 1, 2, . . . , m - 1, j = 1, 2, . . . , m}; (4) e,j(i, m) U Pi-l,o,o-)(m + 1, 2m - i + l), i = 2, 3, . . . , m - 1, j = 1, 2, . . . , (5) :;(I,
m) U Q,(m + l), j = 2, 3, . . . , m;
(6) S:dL m) U Q,(m + 1, 2m);’ (7) P,,j(O, m), i=L 2,. . . , m; (8) {bo, bo v ai, bl v ai}, i = m + 2, m + 3, . . . , 2m + 1. Since Q,(m + 1) > P,,,(m) the set constructed in (6) is a path. Also all the paths are disjoint. The upper covers of [0, b,) are each contained in precisely one path of those in (2), (3), . . . , (8). The upper covers of b. in A0 are upper covers of [0, b,) in 22m+2, these are disposed of in (8). In Al, take the path segments (9) Rj(O, m - l), j = 2, 3, . . . , m;
238
R. Nowakowski
(10) RI@, m - 1)u {YI>u QI<~ + 1, 2~); (11) (Continuation of (5)) Qj(m + 1, 2m), j = 2, 3, . . . , m; (12) (Continuation of part of (7)) P,,,(m). Since {Rj} and {Qj} only intersect {Po,j}, the paths contained in (9)-(12) are mutually disjoint, moreover since {P,,j(l)} = {Rj(l)} every element of L(b,) flAl is contained in precisely one path of (l), (9), (lo), (ll), (12). In A2 take, (13) 9; (14) PO,,; (15) (Continuation (16) {b,
of (7)) PO,j(WZ, 2m + l), j = 2, 3, . . f, m;
bz v em+d;
By the choice of 9 and {Po,j}, all U(b,) n & = bm+z> &I is contained and (16). In A-,, essentially the construction reversed. Specifically, take (17) 8;-{S,jli=l,2,...,m,j=l,2,...,m}; (18) Pi,,i+,G,(i, m - 1) U Pi+l,j(m,
the paths are disjoint and every element of in precisely one path of (2), (13), (14), (15) in Al using the E,j’s, the Q’s and the R’s, is
i = 1, 2, . . . , m - 2,
2m - i - l),
j = 1,
(19) ~~,;;vmiVi,ai),I~i~m-tl,j=l.2,...,m}; (20) (Continuation of (9)) Rj(m - 1) U Pi,j(m, 2m - l), j = 2, 3, . . . , m; (21) R1(O, m - 1) U Pl,l(m, 2m - 1); (22) (Continuation of (15)) Po,j(m + 1, 2m), j = 2, 3, . . . , m; (23) (Continuation of (12)) P,,,(m, 2m). Again, all the paths are disjoint and each element of U(b,) fTA, = [a,,, v u~,,,+~, b3) is contained in precisely one path of (17), (18), (19) and (21). The preceeding construction has produced IL(b,)l may disjoint paths from L(b,) to U(b,) and consequently the extensions of these paths form [L(bJl many disjoint paths from [0, b) to (b, 11.It remains to identify in 22m-c2the families of paths S’, 3’ and B’ corresponding to 2, L?iZand 9 and give the appropriate permutations 0; and nl. The family of paths .Y’={Pk(i=O,l,,.., m,j=1,2 ,..., m+l} with the permutations are pl+l,j+l
=
Ej
V %+l,
4+l(i
+
1)
=
Qi)
fort=~,0,~=1,2 PI+1,* =Pi1vuti+2,
tl(l)=l
Ph, = {b,, bZ, b3}, t;(l)
fort=Jd,
,..., a,i=l,2,.
m-l,j=1,2
,...,
m;
,...,
m;
. . ,m-1;
= 1 for t = zr, a;
P;j= Plj(l, m) U Qj(m + 1) U (Qj(m + 1, 2m) v a,+,),z;(j)= j
forr=n,a,j=2,3
Cutsets of Boolean lattices
P& =
poi(O,
m + 1) V U2m+2)
m) U (POj(m,
j=2,3,.
U(P,i(m+1,2m)VU,+~VU~,+~),
P&,+1 = P,,(O, m) U (P&(m) v u.,+J %
239
U (P,,(m,
. . ,m;
2m) v u,+l v uh+2);
= (&I v uzm+d*
of al and xf are easy to verify for i 3 2 since Pk(m + 1) = Pi_l,j_l(WZ) V U,+l (if j > 1). If i Z m and j = 1 then Pil(m + 1) = Pi_l,l(m) v ati+* which is covered by Pi_l,l(m + 2) and covers Pf_l,l(m). If i = m, then - 1) v u~,,,+~ and Pkl(m + 1) = bI = b. v a&+2. However Pk_l, l(m) = P,_l,l(m P,_l,l(m - 1) is a lower cover of bO, i.e., PLl(m + 1) > Pk_l,l(m). In a similar but dual manner it can be shown that Pk_,,,(m + 2) > PLl(m + 1). The family 9’ is defined as follows. The properties
Qi’ = Qj(‘J, ml U (Qj(m, m + 1) v U
~2~+2)
(Qj(m + 1,2m) v a,+~ v a~~+2),
j = 2,3, . . . , m;
Qi = QI@, 2m) v ~2~+2; Q ;+I = QI<% m - 1) U U
(~~1 U {y2) U {y3)
The family 9’ is given by R; = Q;,
forj=2,3,...,m+l;
R; = (R,(O, 2m) v azrn+2) In particular, note that R, = Ql = PO, so that RI = Q; = P&. Since P,,(m - 1) < yO< P,,(m + l), it follows that P&(m) = P,,(m - 1) v uzmc2-Cy2 -KPol(m + 1) v U 2m+2=P&(m+2). Moreover, for j=2,3,. . . , m, the following covering relations hold: Qi(m + 2) = Qj(m + 1)
v uzrn+2)
Qj(m + 1)
= PQ(m + 1) > Q,(m) = R;(m);
Qi
y3 ) y1 = PL+dm
v em+2 u2m+2= R;(m);
+ 1) > y. =
QL+l(m).
It is easy to verify that the paths from 9’, respectively 9?.‘, each intersect a unique path of Phj. The required families 2’, 9’ and 9’ have been found, together with the appropriate covering relations. Consequently Lemma 6 has been proved and hence so has Theorem 3. 0
240
R. Nowakowski
References [l] M. Aigner, Combinatorial Theory (Springer, New York, 1979). [2] M. Bell and J. Ginsburg, Compact spaces and spaces of maximal complete subgraphs, Trans. Amer. Math. Sot. 283 (1984) 329-338. [3] J. Ginsburg, Compactness and subsets of ordered sets that meet all maximal chains, Order 1 (1984) 147-1.57. [4] J. Ginsburg, I. Rival and B. Sands, Antichains and finite sets that meet all maximal chains, to appear. [5] P.A. Grillet, Maximal chains and antichains, Fund. Math. 65 (1969) 157-167. [6] I. Rival and N. Zaguia, Crosscuts and cutsets, Order 1 (1984) 235-247. [7] N. Sauer and R. Woodrow, Finate cutsets and finite antichains, Order 1 (1984) 35-46.