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Box-Threshold Graphs
Chapter 10 Box-Threshold Graphs 10.1
Introduction
We recall from Condition 5 of Theorem 1.2.4 that the threshold graphs are characterized by the absence of incomparable vertices under the vicinal preorder. The box-threshold graphs form a larger class of graphs in which incomparable vertices are allowed under a certain condition. We denote the fact that vertices x and y are incomparable under the vicinal preorder by x I1 Y, and the fact that vertex x is larger than vertex y under the vicinal preorder by x ~- y. D e f i n i t i o n 10.1.1 A graph G is called a b o x - t h r e s h o l d ( B T ) graph when every two vertices x, y of G satisfy x
Ily~degx=degy,
(10.1)
or equivalently degx>degy~x~-y.
(10.2)
The boxes of a graphs are defined as the blocks of its degree partition, i.e., the equivalence classes of vertices under equality of degrees. Thus Definition 10.1 requires that incomparable vertices belong to the same box. The analogy between BT and threshold graphs will become clear below: it turns out that if every box of a BT graph is shrunk into a single vertex, a threshold graph results. Figure 10.1 exhibits the adjacency matrix of a BT graph, with rows and columns sorted by vertex-degrees. Note that the only incomparable pairs are the two vertices of degree 4 and the two vertices of degree 3. 257
258
Box-Threshold Graphs
Figure 10.1" The adjacency matrix of a BT graph. 116544331 1 1
1
1
1
1
1
1
1
1
1
1
1
1 1 1
Condition 10.2 was introduced by Rawlinson and Entringer IRE79], who applied it to a problem of a communications network. They also found the minimum number of edges of a BT graph with given lower bounds for its degrees. The topic was developed further by Peled and Simeone [PS84], on whose work this section is based. Further work on the subject was done by R.I. Tyshkevich and A.A. Chernyak [TC85a], who used the decomposition method to enumerate the BT graphs, as reported in Chapter 17. In Section 10.2 we present elementary properties of BT graphs. Section 10.3 introduces a model of a symmetric transportation network with priority constraints to characterize the degree sequences of BT graphs. Section 10.4 introduces the concept of the frame of a graph to capture the box interconnection structure, and exhibits the close analogy between the frames of BT graphs and threshold graphs.
10.2
Elementary Properties
It follows immediately from (10.1) that every regular graph is BT, and from 10.2 that adding or deleting an isolated vertex to a BT graph results in a new BT graph. Rawlinson and Entringer [RE79] actually considered only connected BT graphs, but the following proposition shows that every nonconnected BT graph results from adding isolated vertices to some connected BT graph or to some regular graph. A
10.2.1 Let G be a B T graph and let G be obtained from G by deleting its isolated vertices. Then G is connected or regular.
Proposition
10.2
Elementary Properties
259 A
Proof. Since all connectedcomponents of G have edges, every two vertices in different components of G are incomparable. Since G is BT, it follows that every two vertices in different components of G have equal degrees. Therefore G is connected or regular. 9 In Section 11.3 we define matrogenic graphs and characterize them by the absence of the configuration 9C(A, B, C, D, E) consisting of vertices A, B, C, D and E, edges A B , A C and ED, and nonedges E B , E C and AD. The following result is then immediate.
Proposition 10.2.2 The class of B T graphs properly contains that of matrogenic graphs. Proof. By Definition 10.1.1, if G is not BT, then it contains incomparable vertices A and E with deg(A) > deg(E). By the incomparability, there exist vertices B and D such that A B and E D are edges and AD and E B are nonedges. Since deg(A) > deg(E), there e x i s t s a vertex C such t h a t AC is an edge and E C is a nonedge. Thus G contains the configuration .T'(A,B, C , D , E ) and is not matrogenic. An example of a non-matrogenic BT graph is illustrated in Figure 10.2. 9 Figure 10.2: A non-matrogenic BT graph.
I I I I The following two results for BT graphs generalize similar results for threshold graphs.
Proposition 10.2.3 The complement of a B T graph is BT. Proof. Complementation does not alter vertex incomparabilities or equality of degrees. 9
Theorem 10.2.4 B T is a forcible property of a degree sequence. In other words, if G and H are graphs with the same degree sequence and G is BT, then H is also BT. We may thus unambiguously define a degree sequence to be B T if (all) its realizations are B T graphs.
Box-Threshold Graphs
260
P r o o f . Since there is a degree-preserving bijection between the vertices of G and H, we may assume that G and H have the same vertices and that each vertex has equal degrees in G and H. Then by Corollary 3.1.11, H can be obtained from G by a sequence of transfers, where a transfer involves dropping the two edges xu and zy and adding the two nonedges x z and yu of an alternating 4-cycle. We may therefore assume that H is obtained from G by a single transfer as above. Moreover, by the symmetry of the four vertices of the alternating 4-cycle, it is enough to show that if x II P in H, where p is a fifth vertex, then deg(x) = deg(p). Since x II p in H, x and p are opposite corners of an alternating 4-cycle A in H. If neither x u nor x z is a side of A, then A is also an alternating 4-cycle in G and the conclusion follows since G is BT. If both o f x u and x z are sides of A, then x II Y II p i n G, and hence deg(x) = deg(y) = deg(p). If xu is a side of A and x z is not, then the configuration in G and H is as illustrated in Figure 10.3, where the vertices of A are x, u, p, q. Figure 10.3" Configurations in the proof of Theorem 10.2.4. q
x
z
q
T
x
9
i I I
9
I I I
I
I
p
u G
y
I I I
I
i
~k
.L
v
v
p
u
y
9 v
9
I I I
v
,k v
z
H
If pz is an edge, then x I] P in G; otherwise x II Y It P in G. Therefore the conclusion follows in both cases as before. Finally, if xz is a side of A and x u is not, we repeat the argument with pu replacing pz. 9 The next theorem gives an easy test of the BT property. To state it, we use the following definitions. D e f i n i t i o n 10.2.5 A bipartite graph with bipartition ( X , Y ) is said to be c o l o r - r e g u l a r ( C R ) if all the vertices of X have the same degree and all the vertices of Y have the same degree. Two disjoint sets X and Y of vertices in a graph G are said to be c o m p l e t e l y c o n n e c t e d , C R c o n n e c t e d , or
10.2
Elementary Properties
261
c o m p l e t e l y d i s c o n n e c t e d when the edges joining them in G form a com-
plete, a CR, or an edgeless bipartite graph, respectively. A similar definition applies in the case of a connection of X to itself, where the subgraph induced by X should be complete, regular, or edgeless, respectively. Note that complete connection and complete disconnection are special cases of CR connection. D e f i n i t i o n 10.2.6 Assume that a graph G has the degree sequence
d? i.e., exactly m i vertices have degree di, with d l > of degree di constitute the i - t h b o x Bi of G.
"'" > d~. The rni vertices
is B T if and only if for each subscript i there is some k(i) such that Bi is completely connected to B I , . . . ,Bk(i)_l, Ct~ connected to Bk(i), and completely disconnected from
Theorem
Bk(0+l,.
10.2.7 A graph with boxes B I , . . . , B ~
9 9B~.
P r o o f . " O n l y if"" Recall that N(x) denotes the set of neighbors of a vertex x. For every vertex x C B~, let k(z) be the largest k such that N ( z ) N Bk r ~. Then by the BT property (10.2), {x} is completely connected to B 1 , . . . , Bk(x)-I and completely disconnected from Bk(x)+l,..., B~. Moreover, if y is any other vertex of B~, then since deg(z) = deg(y), we must have k(x) = k(y) and IN(x) N Bk(~)l = IN(y)N Bk(y)l. This means, first, that k is a function of i alone and not of the particular x in Bi, and may be denoted by k(i); and, second, that for j = k(i) and trivially for each j 5r k(i) all the vertices x E Bi have the same IN(x)N Bj]. In particular, all the vertices x C Bk(~) have the same IN(z)N Bil. Therefore Bi and Bk(~) are CR connected. " I f " : Let x II Y for some x C B~, y E Bj. We have to show that i = j. First of all, k(i) = k(j), for if, say, k(i) > k(j), then N(y) C_ N(z), contradicting non-comparability. Let us denote by p the common value of k(i) and k(j). The only way that x and y can be non-comparable is for each of them to have a neighbor in Bp that is not a neighbor of the other. Therefore Bi and Bj are neither completely connected to Bp nor completely disconnected from Bp. This implies that k(p) is equal to both i and j. " Note that a BT graph is a threshold graph if and only if every two boxes are either completely connected or completely disconnected. This means that
Box-Threshold Graphs
262
the adjacency matrix is equal to the corrected Ferrers diagram of the degree sequence. Deleting a vertex of degree 1 from the graph of Figure 10.2 results in a non-BT graph. This shows that BT is not hereditary, and hence cannot be characterized by forbidden configurations. However, the next theorem shows that the BT property is preserved when an entire box is deleted from the graph. The results of Section 10.4 can be interpreted as a characterization of the BT property in terms of forbidden configurations of boxes. T h e o r e m 10.2.8 If G is BT and H is obtained from G by deleting an entire
box, then H is also BT. P r o o f . let the vertices of the deleted box have degree d in G. Assume that, if possible, H is not BT. Then it has vertices x, y, z such that degu(x ) > degH(y), yet z is adjacent to y but not to x. Since G has the same configuration and is BT, dega(x ) _< dega(y). Hence d e g a ( y ) - degH(y) > dega(x ) - d e g u ( x ) , which means that in G, Y has more neighbors of degree d than x has. By Theorem 10.2.7, y is adjacent in G and H at least to all vertices u such that dega(u) > d, whereas x is adjacent in H at most to these vertices. Hence degu(y ) > degH(x), which is a contradiction. ..
10.3
A Transportation M o d e l
Theorem 10.2.4 raises the question of characterizing BT sequences without constructing an arbitrary realization and testing it for the BT property. Here we give such a characterization, which does not even assume that the sequence is realizable. A transportation network consists of suppliers A1,...,A~ with supplies al,. 9 a~ of some commodity, consumers B 1 , . . . , B~ with demands b l , . . . , b~, and capacities c~j of the route from A~ to Bj. A flow is a matrix (f~j) such that 0 <_ f~j <_ c~j. We say that the flow f satisfies the supplies (resp. demands) when it satisfies the equations Ej f i j -- ai (resp. Ei f i j -- a j ) . We define a special flow r called the supplier-priority flow, by the following recursion: ~)ij --
min { ai
jl
-- E
}
r
cij
9
(10.3)
k=l
This has the interpretation that each supplier A i sends as much as possible to the first consumer B1, limited only by its own supply ai and the capacity eil,
10.3
A Transportation Model
263
regardless of the demand bl; then to B2, limited only by the residual supply and the capacity ci2; and so on. Clearly, under the feasibility conditions ~-'~j cij > ai, ~ satisfies the supplies. Similarly, we define the consumerpriority flow ~ by the recursion
r
- min {bj - ~ ~zkj, cij} ,
(10.4)
k=l
which satisfies the demands under the feasibility conditions ~ i cij > bj. The network is said to be symmetric when r = s, ai = bi for all i, and cij = cji for all i, j. T h e o r e m 10.3.1 In a symmetric transportation network satisfying the feasibility conditions, the following conditions are equivalent:
1. r satisfies the d e m a n d s ; 2. r
= Cji for all i, j;
3. r 1 6 2 P r o o f . 1) =~ 2): Assume that r < Cji for some i,j. Since Cji <_ cji = cij, we have r cij, and hence by (10.3) elk = 0 for all k > j. Similarly, since 0 < Cji, we have Cki = cki = cik for all k < j. Therefore
bi
=ai
=
k
k
k
k
and so r does not satisfy the demands. 2) =~ 3): We show by induction on i that r r
=
min{bj, Clj }
--
If ~kj = Ckj holds for all k < i, then
min{
k=l
k
~)ij. For i = 1 we have
=
min{aj, Cjl } = Cjl = r
min{a
=
=
k=l
3) =~ 1): This is obvious, since r satisfies the demands. W i t h every sequence d l 1.-- d~mr, dl > ... > dr, of non-negative integers we associate a symmetric transportation network given by
ai = bi = dimi,
cij = m i ( m j - ~ij),
264
Box-Threshold Graphs
where
1, i f i - j O, if not.
5ij -
T h e o r e m 10.3.2 A sequence d~ ~ . . . d ~ ~, dl :> "" :> dr, of non-negative integers is B T if and only if dl < ~ i mi and the associated symmetric transportation networks is such that the conditions of Theorem 10.3.1 hold and the eli are even integers. Before proving the theorem, let us remark that the "if" part does not assume that the sequence is graphical. Another remark is that the condition dl < ~-~imi is equivalent to the feasibility condition ~ j c~j = m ~ ( ~ j m j - 1 ) _> midi = ai, and thus guarantees the equivalence of the three conditions of Theorem 10.3.1. An example of a BT degree sequence is (23)2(20)4(13)3(11)56446. The vector (ai) is (46, 80, 39, 55, 24, 24) and the matrices (cij) and (r are shown in Figure 10.4 Figure 10.4: Capacity and flow matrices for the sequence (23)2(20)4(13)3(11)~6446.
(Cij) --
2 8 6 10 8 12
8 12 12 20 16 24
6 12 6 15 12 18
10 20 15 20 20 30
8 16 12 20 12 24
12 24 18 30 24 30
2 8 6 10 8 12
(r
8 6 10 8 12 12 12 20 16 12 12 6 15 0 0 20 15 0 0 0 16 0 0 0 0 12 0 0 0 0
P r o o f . To prove the necessity, assume that the BT graph G has degree sequence d~ 1 . . . d~ ~. Clearly dl < ~-,i mi, because G has ~-~i mi vertices in total. To prove the other conditions, let xij be the number of edges connecting Bi and Bj for i r j, and let Xii be twice the number of edges within Bi. Thus xij = xji and Xii is even. Moreover, ai = midi is the sum of the degrees of the vertices of Bi, and cij = r n i ( m j - (~ij) is the upper bound for xij reflecting the absence of multiple edges and loops. Hence by Theorem 10.2.7, j-1 X ij --
min
ai-
E
} X ik, Cij
-- (~ij.
k=l
To prove the sufficiency we need the following two facts.
10.4
Frames of BT Graphs
265
F a c t 1 If f is a common multiple of the positive integers p and q and f <_ pq, then there exists a biregular graph with degree sequence ( f /p)V(f /q)q. P r o o f . To construct such a graph, take p vertices x 0 , . . . , Xp_l and q vertices yo,...,yq-1, put r - f / p , and join x0 with y 0 , . . . , y ~ - l , Xl with y~,...,y2~_l, and so on in a cyclic fashion, where the indices are taken modulo q. This gives to each xi the degree r for a total of f edges. Since f is divisible by q, it also gives to each yj the same degree f / q . This proves Fact 1. F a c t 2 If f is an even integer divisible by the positive integer p and f <_ p ( p - 1), then there exists a d-regular graph on p vertices, where d - f /p. P r o o f . Note that d _< p - 1 by assumption. We distinguish the cases p even and p odd. For p even, the complete bipartite graph I(p/2,p/2 is a union of p/2 edgedisjoint perfect matchings (if the vertices are x0,. 9 xp/2-1 and y0,. 9 Y p / 2 - 1 , then the i-th matching joins xy with yj+i with indices modulo p/2). If d _< p/2, take the union of any d matchings, and if d > p/2, take the complement (with respect to Kp) of the union of any p - 1 - d matchings. For p odd, d must be even as f is even. Take p vertices x 0 , . . . , xp-1 and join xi with X i • Xi+d/2. This proves Fact 2. To complete the proof of Theorem 10.3.2, we show that the conditions are sufficient by constructing a BT graph with the given degree sequence. Put mi vertices in cell Ci for each i. Since ai and cij are multiples of mi, so is r For every i r j, since r - Cji, this number is a common multiple of mi and rnj, and furthermore r <_ cij - mimj. Therefore by Fact 1, we can construct a biregular connection between Ci and Cj such that each vertex of Ci has r neighbors in Cj and each vertex of Cj has r neighbors in Ci. Similarly for each i, r iS an even integer divisible by rag, and r <_ ci~ - m i ( m ~ - 1). Therefore by Fact 2, we can construct a (r regular graph on Ci. In the resulting graph G, all the m i vertices of Ci have the same degree, and the sum of these m i degrees is ~ j r - a i - m i d i because r satisfies the supplies. Thus the boxes of G are the Ci and its degree sequence is d~n l . - . d m~. Finally, G is BT by Theorem 10.2.7. 9
10.4
F r a m e s of B T G r a p h s
In order to represent the box-interconnection structure of a graph we use the concept of a frame. The frame of a graph G can be thought of as a graph obtained by shrinking each box of G into a single vertex and coloring
266
Box-Threshold Graphs
the resulting edges to indicate whether or not the boxes were completely connected. Definition 10.4.1 Let G be a graph whose boxes are BI , . . . , Br. The f r a m e of G is a graph F on { B I , . . . , B~} in which Bi and Bj are joined by a black edge, a red edge, or no edge if they are completely connected, partially but not completely connected, or completely disconnected in G, respectively. This includes the case that i - j , and so F has black and red loops. In the case that IBil - 1, there is an ambiguity whether Bi should have a black loop or no loop, since Bi is both completely connected to itself and completely disconnected from itself in G. We resolve this ambiguity by giving to Bi a black loop in F when Bi is connected in G to some box Bk with i > k, i.e., when some vertex of Bi has a neighbor with smaller degree in G. The set of all black (red) edges and loops is denoted by B (R). We denote the box containing a vertex x by 2. In this section we characterize when G is BT in terms of its frame F, and also characterize when a given F is the frame of some BT graph G. For this we use the following definition. Definition 10.4.2 A m a t c h i n g w i t h loops is an undirected graph with loops, no two of whose edges are incident to the same vertex, i.e., its adjacency matrix does not have two 1 's in the same row or column. A t h r e s h o l d g r a p h w i t h loops is an undirected graph with loops such that the rows (and columns) of its adjacency matrix are totally ordered by set-inclusion. An equivalent definition is that it is the underlying graph with loops of a symmetric Ferrets digraph (Section 2.1). If the loops are removed from a threshold graph with loops, it becomes a threshold graph by Theorem 2.1.3, and every threshold graph can be obtained in this way by Theorem 2.1.~. T h e o r e m 10.4.3 Let F be the frame of G. Then G is B T if and only if 1. F is a threshold graph with loops; 2. the black edges of F form a threshold graph with loops; 3. the red edges of F form a matching with loops.
P r o o f . " O n l y if"" The adjacency matrix of F is obtained from the flow matrix r upon replacing r by
10.4
F r a m e s of B T G r a p h s
9 a b l a c k 1 if r
-
cij
> 0 or if
267
i -
j,
eli
-
cii -
0
and ~bik > 0 for some
k>i; 9 ared 1 if 0 < r
< Cij;
9 a 0in all other cases. Since (r is symmetric by Theorem 10.3.2, and by the definition (10.3) of r F satisfies Conditions 1 through 3. "If"" We assume that deg a x > deg a y and show that z ~ y in G. Some third vertex z is adjacent to z but not to y. We assert that for each vertex w of G, i f t ~ C B, then wz C B. Indeed, 22 C B U R and 2~ ~ B. If zx C B, then t~2 C B by Condition 2 as asserted. If 22 C R, then 29 ~ R by Condition 3, i.e., 2~ is not an edge of F, hence xw C B U R by Condition 1. But t~ 7~ 2 because t ~ is an edge of F and 2~ is not, hence xw ~ R by Condition 3, and therefore 2t~ C B as asserted. Returning to the proof that x ~- y, we assume that, if possible, w r x, y, wy is an edge of G and wx is not. Then t~2 ~ B, so t ~ ~ R by the assertion, hence wy C R. Therefore t~2 ~ R by Condition 3, i.e., t~2 is not an edge of F. Since zx is an edge of F, we have 2 :/- t~, and since wy C R, we have zy ~ R by Condition 3. Hence zy C B by Condition 1, and this contradicts the fact that z is not adjacent to y in G. " T h e o r e m 10.4.4 Let F be a graph with loops whose edges are colored black or red. Then F with its colors is the frame of some B T graph if and only if 1. F is a threshold graph with loops; 2. the black edges of F form a threshold graph with loops; 3. the red edges of F form a matching with loops; ~. every two vertices of F differ in their number of incident black edges or in their number of incident red edges. We say that the vertices differ in their black degree or their red degree, where a loop contributes 1 to the degree. Moreover, in that case no three vertices of F can have the same black degree.
268
Box-Threshold Graphs
P r o o f . " O n l y if"" Assume that F is the frame of a BT graph G. By Theorem 10.4.3, Conditions 1 through 3 hold. To prove Condition 4, assume that ~ and ~ are distinct vertices of F, say deg a z > deg a y, and that ~ and have the same black degree. By Condition 2, for every z, z z E B if and only if ~2 E B. But some z is adjacent to x and not to y in G, hence zy ~ B and 22 E R. By Condition 3, ~,~ ~ R, i.e., 29 is not an edge of F. We assert that no red edge is incident with 9, which shows that 2 and 9 differ in their red degrees. Indeed, if @~ E R, then w x ~ R by Condition 3 and @2 ~ B by the above, and this contradicts Condition 1, thus proving t h e assertion and Condition 4. Moreover, if 2 and r have the same black degree, then by Conditions 3 and 4, their red degrees differ by exactly 1, and so no three vertices can have the same black degree. "If"" We construct a BT graph G whose frame is F. For each vertex v of F, let G have four vertices v0, vl, v2, v3. For each black edge v w E t3 of F with v 7~ w, join each vi to each wj. For each red edge v w E R of F with v -~ w, join each vi to wi and Wi+l (indices modulo 4). For each black loop vv E B of F, join each vi to each vj, i 7~ j . For each red loop vv E R of F, join each Vi to Vi+I and v i - , . This construction is illustrated in Figure 10.5. Note that each red edge or loop of F contributes 2 to the degrees of the vertices of G corresponding to its ends, a black loop contributes 3, and a black edge that is not a loop contributes 4. It is sufficient to show that F is the frame of G, for this implies that G is BT by Theorem 10.4.3. Clearly for each vertex v of F, all the vi have the same degree in G. Therefore it only remains to show that if v and w are distinct vertices of F, then deg a v0 =fi dega w0. If v and w have the same black degree, then by Condition 2, either both have a black loop or both do not, so the black edges incident to v and w contribute equally to the degrees of v0 and w0. Also by Condition 4, v and w have different red degrees, hence deg a v0 r deg a w0 as required. Assume now that the black degree of v is larger than the black degree of w. Then by Condition 2, for each vertex x of F, z w E B implies x v E B , but not conversely. Therefore the black edges contribute more to deg a v0 than to dega w0. We now prove that for each vertex y of F, y w E R implies yv E R U B , which shows that deg C v0 > dega w0 as required. Assume that, if possible, y w E R and yv ~ R U B. Let x be a vertex of F such that x v E B and x w ~ B. Then x-~ y (sinceyv ~ B ) , hence by Condition 3, x w ~ R, i.e., x w is not an edge of F. This contradicts Condition 1. .. The construction used in the "if" part of the proof of Theorem 10.4.4 can be generalized to yield all the degree sequences of BT graphs with a given frame.
10.4
Frames of BT Graphs
269
Figure 10.5: The construction of a BT graph with a given frame.
vo . . . . ~
wo
V0
W0
Vl
Wl
-
Wl
V2 ~
W2
.,.
W2
V3 -
. W3
?-)3
.,.
vwEB
W3
vwER
V2
"
-
131
V2
"
V3
...
-
VO
Y3
.
vv E B
~
Vl
-
vv E R
VO
Introduction
We recall from Condition 5 of Theorem 1.2.4 that the threshold graphs are characterized by the absence of incomparable vertices under the vicinal preorder. The box-threshold graphs form a larger class of graphs in which incomparable vertices are allowed under a certain condition. We denote the fact that vertices x and y are incomparable under the vicinal preorder by x I1 Y, and the fact that vertex x is larger than vertex y under the vicinal preorder by x ~- y. D e f i n i t i o n 10.1.1 A graph G is called a b o x - t h r e s h o l d ( B T ) graph when every two vertices x, y of G satisfy x
Ily~degx=degy,
(10.1)
or equivalently degx>degy~x~-y.
(10.2)
The boxes of a graphs are defined as the blocks of its degree partition, i.e., the equivalence classes of vertices under equality of degrees. Thus Definition 10.1 requires that incomparable vertices belong to the same box. The analogy between BT and threshold graphs will become clear below: it turns out that if every box of a BT graph is shrunk into a single vertex, a threshold graph results. Figure 10.1 exhibits the adjacency matrix of a BT graph, with rows and columns sorted by vertex-degrees. Note that the only incomparable pairs are the two vertices of degree 4 and the two vertices of degree 3. 257
258
Box-Threshold Graphs
Figure 10.1" The adjacency matrix of a BT graph. 116544331 1 1
1
1
1
1
1
1
1
1
1
1
1
1 1 1
Condition 10.2 was introduced by Rawlinson and Entringer IRE79], who applied it to a problem of a communications network. They also found the minimum number of edges of a BT graph with given lower bounds for its degrees. The topic was developed further by Peled and Simeone [PS84], on whose work this section is based. Further work on the subject was done by R.I. Tyshkevich and A.A. Chernyak [TC85a], who used the decomposition method to enumerate the BT graphs, as reported in Chapter 17. In Section 10.2 we present elementary properties of BT graphs. Section 10.3 introduces a model of a symmetric transportation network with priority constraints to characterize the degree sequences of BT graphs. Section 10.4 introduces the concept of the frame of a graph to capture the box interconnection structure, and exhibits the close analogy between the frames of BT graphs and threshold graphs.
10.2
Elementary Properties
It follows immediately from (10.1) that every regular graph is BT, and from 10.2 that adding or deleting an isolated vertex to a BT graph results in a new BT graph. Rawlinson and Entringer [RE79] actually considered only connected BT graphs, but the following proposition shows that every nonconnected BT graph results from adding isolated vertices to some connected BT graph or to some regular graph. A
10.2.1 Let G be a B T graph and let G be obtained from G by deleting its isolated vertices. Then G is connected or regular.
Proposition
10.2
Elementary Properties
259 A
Proof. Since all connectedcomponents of G have edges, every two vertices in different components of G are incomparable. Since G is BT, it follows that every two vertices in different components of G have equal degrees. Therefore G is connected or regular. 9 In Section 11.3 we define matrogenic graphs and characterize them by the absence of the configuration 9C(A, B, C, D, E) consisting of vertices A, B, C, D and E, edges A B , A C and ED, and nonedges E B , E C and AD. The following result is then immediate.
Proposition 10.2.2 The class of B T graphs properly contains that of matrogenic graphs. Proof. By Definition 10.1.1, if G is not BT, then it contains incomparable vertices A and E with deg(A) > deg(E). By the incomparability, there exist vertices B and D such that A B and E D are edges and AD and E B are nonedges. Since deg(A) > deg(E), there e x i s t s a vertex C such t h a t AC is an edge and E C is a nonedge. Thus G contains the configuration .T'(A,B, C , D , E ) and is not matrogenic. An example of a non-matrogenic BT graph is illustrated in Figure 10.2. 9 Figure 10.2: A non-matrogenic BT graph.
I I I I The following two results for BT graphs generalize similar results for threshold graphs.
Proposition 10.2.3 The complement of a B T graph is BT. Proof. Complementation does not alter vertex incomparabilities or equality of degrees. 9
Theorem 10.2.4 B T is a forcible property of a degree sequence. In other words, if G and H are graphs with the same degree sequence and G is BT, then H is also BT. We may thus unambiguously define a degree sequence to be B T if (all) its realizations are B T graphs.
Box-Threshold Graphs
260
P r o o f . Since there is a degree-preserving bijection between the vertices of G and H, we may assume that G and H have the same vertices and that each vertex has equal degrees in G and H. Then by Corollary 3.1.11, H can be obtained from G by a sequence of transfers, where a transfer involves dropping the two edges xu and zy and adding the two nonedges x z and yu of an alternating 4-cycle. We may therefore assume that H is obtained from G by a single transfer as above. Moreover, by the symmetry of the four vertices of the alternating 4-cycle, it is enough to show that if x II P in H, where p is a fifth vertex, then deg(x) = deg(p). Since x II p in H, x and p are opposite corners of an alternating 4-cycle A in H. If neither x u nor x z is a side of A, then A is also an alternating 4-cycle in G and the conclusion follows since G is BT. If both o f x u and x z are sides of A, then x II Y II p i n G, and hence deg(x) = deg(y) = deg(p). If xu is a side of A and x z is not, then the configuration in G and H is as illustrated in Figure 10.3, where the vertices of A are x, u, p, q. Figure 10.3" Configurations in the proof of Theorem 10.2.4. q
x
z
q
T
x
9
i I I
9
I I I
I
I
p
u G
y
I I I
I
i
~k
.L
v
v
p
u
y
9 v
9
I I I
v
,k v
z
H
If pz is an edge, then x I] P in G; otherwise x II Y It P in G. Therefore the conclusion follows in both cases as before. Finally, if xz is a side of A and x u is not, we repeat the argument with pu replacing pz. 9 The next theorem gives an easy test of the BT property. To state it, we use the following definitions. D e f i n i t i o n 10.2.5 A bipartite graph with bipartition ( X , Y ) is said to be c o l o r - r e g u l a r ( C R ) if all the vertices of X have the same degree and all the vertices of Y have the same degree. Two disjoint sets X and Y of vertices in a graph G are said to be c o m p l e t e l y c o n n e c t e d , C R c o n n e c t e d , or
10.2
Elementary Properties
261
c o m p l e t e l y d i s c o n n e c t e d when the edges joining them in G form a com-
plete, a CR, or an edgeless bipartite graph, respectively. A similar definition applies in the case of a connection of X to itself, where the subgraph induced by X should be complete, regular, or edgeless, respectively. Note that complete connection and complete disconnection are special cases of CR connection. D e f i n i t i o n 10.2.6 Assume that a graph G has the degree sequence
d? i.e., exactly m i vertices have degree di, with d l > of degree di constitute the i - t h b o x Bi of G.
"'" > d~. The rni vertices
is B T if and only if for each subscript i there is some k(i) such that Bi is completely connected to B I , . . . ,Bk(i)_l, Ct~ connected to Bk(i), and completely disconnected from
Theorem
Bk(0+l,.
10.2.7 A graph with boxes B I , . . . , B ~
9 9B~.
P r o o f . " O n l y if"" Recall that N(x) denotes the set of neighbors of a vertex x. For every vertex x C B~, let k(z) be the largest k such that N ( z ) N Bk r ~. Then by the BT property (10.2), {x} is completely connected to B 1 , . . . , Bk(x)-I and completely disconnected from Bk(x)+l,..., B~. Moreover, if y is any other vertex of B~, then since deg(z) = deg(y), we must have k(x) = k(y) and IN(x) N Bk(~)l = IN(y)N Bk(y)l. This means, first, that k is a function of i alone and not of the particular x in Bi, and may be denoted by k(i); and, second, that for j = k(i) and trivially for each j 5r k(i) all the vertices x E Bi have the same IN(x)N Bj]. In particular, all the vertices x C Bk(~) have the same IN(z)N Bil. Therefore Bi and Bk(~) are CR connected. " I f " : Let x II Y for some x C B~, y E Bj. We have to show that i = j. First of all, k(i) = k(j), for if, say, k(i) > k(j), then N(y) C_ N(z), contradicting non-comparability. Let us denote by p the common value of k(i) and k(j). The only way that x and y can be non-comparable is for each of them to have a neighbor in Bp that is not a neighbor of the other. Therefore Bi and Bj are neither completely connected to Bp nor completely disconnected from Bp. This implies that k(p) is equal to both i and j. " Note that a BT graph is a threshold graph if and only if every two boxes are either completely connected or completely disconnected. This means that
Box-Threshold Graphs
262
the adjacency matrix is equal to the corrected Ferrers diagram of the degree sequence. Deleting a vertex of degree 1 from the graph of Figure 10.2 results in a non-BT graph. This shows that BT is not hereditary, and hence cannot be characterized by forbidden configurations. However, the next theorem shows that the BT property is preserved when an entire box is deleted from the graph. The results of Section 10.4 can be interpreted as a characterization of the BT property in terms of forbidden configurations of boxes. T h e o r e m 10.2.8 If G is BT and H is obtained from G by deleting an entire
box, then H is also BT. P r o o f . let the vertices of the deleted box have degree d in G. Assume that, if possible, H is not BT. Then it has vertices x, y, z such that degu(x ) > degH(y), yet z is adjacent to y but not to x. Since G has the same configuration and is BT, dega(x ) _< dega(y). Hence d e g a ( y ) - degH(y) > dega(x ) - d e g u ( x ) , which means that in G, Y has more neighbors of degree d than x has. By Theorem 10.2.7, y is adjacent in G and H at least to all vertices u such that dega(u) > d, whereas x is adjacent in H at most to these vertices. Hence degu(y ) > degH(x), which is a contradiction. ..
10.3
A Transportation M o d e l
Theorem 10.2.4 raises the question of characterizing BT sequences without constructing an arbitrary realization and testing it for the BT property. Here we give such a characterization, which does not even assume that the sequence is realizable. A transportation network consists of suppliers A1,...,A~ with supplies al,. 9 a~ of some commodity, consumers B 1 , . . . , B~ with demands b l , . . . , b~, and capacities c~j of the route from A~ to Bj. A flow is a matrix (f~j) such that 0 <_ f~j <_ c~j. We say that the flow f satisfies the supplies (resp. demands) when it satisfies the equations Ej f i j -- ai (resp. Ei f i j -- a j ) . We define a special flow r called the supplier-priority flow, by the following recursion: ~)ij --
min { ai
jl
-- E
}
r
cij
9
(10.3)
k=l
This has the interpretation that each supplier A i sends as much as possible to the first consumer B1, limited only by its own supply ai and the capacity eil,
10.3
A Transportation Model
263
regardless of the demand bl; then to B2, limited only by the residual supply and the capacity ci2; and so on. Clearly, under the feasibility conditions ~-'~j cij > ai, ~ satisfies the supplies. Similarly, we define the consumerpriority flow ~ by the recursion
r
- min {bj - ~ ~zkj, cij} ,
(10.4)
k=l
which satisfies the demands under the feasibility conditions ~ i cij > bj. The network is said to be symmetric when r = s, ai = bi for all i, and cij = cji for all i, j. T h e o r e m 10.3.1 In a symmetric transportation network satisfying the feasibility conditions, the following conditions are equivalent:
1. r satisfies the d e m a n d s ; 2. r
= Cji for all i, j;
3. r 1 6 2 P r o o f . 1) =~ 2): Assume that r < Cji for some i,j. Since Cji <_ cji = cij, we have r cij, and hence by (10.3) elk = 0 for all k > j. Similarly, since 0 < Cji, we have Cki = cki = cik for all k < j. Therefore
bi
=ai
=
k
k
k
k
and so r does not satisfy the demands. 2) =~ 3): We show by induction on i that r r
=
min{bj, Clj }
--
If ~kj = Ckj holds for all k < i, then
min{
k=l
k
~)ij. For i = 1 we have
=
min{aj, Cjl } = Cjl = r
min{a
=
=
k=l
3) =~ 1): This is obvious, since r satisfies the demands. W i t h every sequence d l 1.-- d~mr, dl > ... > dr, of non-negative integers we associate a symmetric transportation network given by
ai = bi = dimi,
cij = m i ( m j - ~ij),
264
Box-Threshold Graphs
where
1, i f i - j O, if not.
5ij -
T h e o r e m 10.3.2 A sequence d~ ~ . . . d ~ ~, dl :> "" :> dr, of non-negative integers is B T if and only if dl < ~ i mi and the associated symmetric transportation networks is such that the conditions of Theorem 10.3.1 hold and the eli are even integers. Before proving the theorem, let us remark that the "if" part does not assume that the sequence is graphical. Another remark is that the condition dl < ~-~imi is equivalent to the feasibility condition ~ j c~j = m ~ ( ~ j m j - 1 ) _> midi = ai, and thus guarantees the equivalence of the three conditions of Theorem 10.3.1. An example of a BT degree sequence is (23)2(20)4(13)3(11)56446. The vector (ai) is (46, 80, 39, 55, 24, 24) and the matrices (cij) and (r are shown in Figure 10.4 Figure 10.4: Capacity and flow matrices for the sequence (23)2(20)4(13)3(11)~6446.
(Cij) --
2 8 6 10 8 12
8 12 12 20 16 24
6 12 6 15 12 18
10 20 15 20 20 30
8 16 12 20 12 24
12 24 18 30 24 30
2 8 6 10 8 12
(r
8 6 10 8 12 12 12 20 16 12 12 6 15 0 0 20 15 0 0 0 16 0 0 0 0 12 0 0 0 0
P r o o f . To prove the necessity, assume that the BT graph G has degree sequence d~ 1 . . . d~ ~. Clearly dl < ~-,i mi, because G has ~-~i mi vertices in total. To prove the other conditions, let xij be the number of edges connecting Bi and Bj for i r j, and let Xii be twice the number of edges within Bi. Thus xij = xji and Xii is even. Moreover, ai = midi is the sum of the degrees of the vertices of Bi, and cij = r n i ( m j - (~ij) is the upper bound for xij reflecting the absence of multiple edges and loops. Hence by Theorem 10.2.7, j-1 X ij --
min
ai-
E
} X ik, Cij
-- (~ij.
k=l
To prove the sufficiency we need the following two facts.
10.4
Frames of BT Graphs
265
F a c t 1 If f is a common multiple of the positive integers p and q and f <_ pq, then there exists a biregular graph with degree sequence ( f /p)V(f /q)q. P r o o f . To construct such a graph, take p vertices x 0 , . . . , Xp_l and q vertices yo,...,yq-1, put r - f / p , and join x0 with y 0 , . . . , y ~ - l , Xl with y~,...,y2~_l, and so on in a cyclic fashion, where the indices are taken modulo q. This gives to each xi the degree r for a total of f edges. Since f is divisible by q, it also gives to each yj the same degree f / q . This proves Fact 1. F a c t 2 If f is an even integer divisible by the positive integer p and f <_ p ( p - 1), then there exists a d-regular graph on p vertices, where d - f /p. P r o o f . Note that d _< p - 1 by assumption. We distinguish the cases p even and p odd. For p even, the complete bipartite graph I(p/2,p/2 is a union of p/2 edgedisjoint perfect matchings (if the vertices are x0,. 9 xp/2-1 and y0,. 9 Y p / 2 - 1 , then the i-th matching joins xy with yj+i with indices modulo p/2). If d _< p/2, take the union of any d matchings, and if d > p/2, take the complement (with respect to Kp) of the union of any p - 1 - d matchings. For p odd, d must be even as f is even. Take p vertices x 0 , . . . , xp-1 and join xi with X i • Xi+d/2. This proves Fact 2. To complete the proof of Theorem 10.3.2, we show that the conditions are sufficient by constructing a BT graph with the given degree sequence. Put mi vertices in cell Ci for each i. Since ai and cij are multiples of mi, so is r For every i r j, since r - Cji, this number is a common multiple of mi and rnj, and furthermore r <_ cij - mimj. Therefore by Fact 1, we can construct a biregular connection between Ci and Cj such that each vertex of Ci has r neighbors in Cj and each vertex of Cj has r neighbors in Ci. Similarly for each i, r iS an even integer divisible by rag, and r <_ ci~ - m i ( m ~ - 1). Therefore by Fact 2, we can construct a (r regular graph on Ci. In the resulting graph G, all the m i vertices of Ci have the same degree, and the sum of these m i degrees is ~ j r - a i - m i d i because r satisfies the supplies. Thus the boxes of G are the Ci and its degree sequence is d~n l . - . d m~. Finally, G is BT by Theorem 10.2.7. 9
10.4
F r a m e s of B T G r a p h s
In order to represent the box-interconnection structure of a graph we use the concept of a frame. The frame of a graph G can be thought of as a graph obtained by shrinking each box of G into a single vertex and coloring
266
Box-Threshold Graphs
the resulting edges to indicate whether or not the boxes were completely connected. Definition 10.4.1 Let G be a graph whose boxes are BI , . . . , Br. The f r a m e of G is a graph F on { B I , . . . , B~} in which Bi and Bj are joined by a black edge, a red edge, or no edge if they are completely connected, partially but not completely connected, or completely disconnected in G, respectively. This includes the case that i - j , and so F has black and red loops. In the case that IBil - 1, there is an ambiguity whether Bi should have a black loop or no loop, since Bi is both completely connected to itself and completely disconnected from itself in G. We resolve this ambiguity by giving to Bi a black loop in F when Bi is connected in G to some box Bk with i > k, i.e., when some vertex of Bi has a neighbor with smaller degree in G. The set of all black (red) edges and loops is denoted by B (R). We denote the box containing a vertex x by 2. In this section we characterize when G is BT in terms of its frame F, and also characterize when a given F is the frame of some BT graph G. For this we use the following definition. Definition 10.4.2 A m a t c h i n g w i t h loops is an undirected graph with loops, no two of whose edges are incident to the same vertex, i.e., its adjacency matrix does not have two 1 's in the same row or column. A t h r e s h o l d g r a p h w i t h loops is an undirected graph with loops such that the rows (and columns) of its adjacency matrix are totally ordered by set-inclusion. An equivalent definition is that it is the underlying graph with loops of a symmetric Ferrets digraph (Section 2.1). If the loops are removed from a threshold graph with loops, it becomes a threshold graph by Theorem 2.1.3, and every threshold graph can be obtained in this way by Theorem 2.1.~. T h e o r e m 10.4.3 Let F be the frame of G. Then G is B T if and only if 1. F is a threshold graph with loops; 2. the black edges of F form a threshold graph with loops; 3. the red edges of F form a matching with loops.
P r o o f . " O n l y if"" The adjacency matrix of F is obtained from the flow matrix r upon replacing r by
10.4
F r a m e s of B T G r a p h s
9 a b l a c k 1 if r
-
cij
> 0 or if
267
i -
j,
eli
-
cii -
0
and ~bik > 0 for some
k>i; 9 ared 1 if 0 < r
< Cij;
9 a 0in all other cases. Since (r is symmetric by Theorem 10.3.2, and by the definition (10.3) of r F satisfies Conditions 1 through 3. "If"" We assume that deg a x > deg a y and show that z ~ y in G. Some third vertex z is adjacent to z but not to y. We assert that for each vertex w of G, i f t ~ C B, then wz C B. Indeed, 22 C B U R and 2~ ~ B. If zx C B, then t~2 C B by Condition 2 as asserted. If 22 C R, then 29 ~ R by Condition 3, i.e., 2~ is not an edge of F, hence xw C B U R by Condition 1. But t~ 7~ 2 because t ~ is an edge of F and 2~ is not, hence xw ~ R by Condition 3, and therefore 2t~ C B as asserted. Returning to the proof that x ~- y, we assume that, if possible, w r x, y, wy is an edge of G and wx is not. Then t~2 ~ B, so t ~ ~ R by the assertion, hence wy C R. Therefore t~2 ~ R by Condition 3, i.e., t~2 is not an edge of F. Since zx is an edge of F, we have 2 :/- t~, and since wy C R, we have zy ~ R by Condition 3. Hence zy C B by Condition 1, and this contradicts the fact that z is not adjacent to y in G. " T h e o r e m 10.4.4 Let F be a graph with loops whose edges are colored black or red. Then F with its colors is the frame of some B T graph if and only if 1. F is a threshold graph with loops; 2. the black edges of F form a threshold graph with loops; 3. the red edges of F form a matching with loops; ~. every two vertices of F differ in their number of incident black edges or in their number of incident red edges. We say that the vertices differ in their black degree or their red degree, where a loop contributes 1 to the degree. Moreover, in that case no three vertices of F can have the same black degree.
268
Box-Threshold Graphs
P r o o f . " O n l y if"" Assume that F is the frame of a BT graph G. By Theorem 10.4.3, Conditions 1 through 3 hold. To prove Condition 4, assume that ~ and ~ are distinct vertices of F, say deg a z > deg a y, and that ~ and have the same black degree. By Condition 2, for every z, z z E B if and only if ~2 E B. But some z is adjacent to x and not to y in G, hence zy ~ B and 22 E R. By Condition 3, ~,~ ~ R, i.e., 29 is not an edge of F. We assert that no red edge is incident with 9, which shows that 2 and 9 differ in their red degrees. Indeed, if @~ E R, then w x ~ R by Condition 3 and @2 ~ B by the above, and this contradicts Condition 1, thus proving t h e assertion and Condition 4. Moreover, if 2 and r have the same black degree, then by Conditions 3 and 4, their red degrees differ by exactly 1, and so no three vertices can have the same black degree. "If"" We construct a BT graph G whose frame is F. For each vertex v of F, let G have four vertices v0, vl, v2, v3. For each black edge v w E t3 of F with v 7~ w, join each vi to each wj. For each red edge v w E R of F with v -~ w, join each vi to wi and Wi+l (indices modulo 4). For each black loop vv E B of F, join each vi to each vj, i 7~ j . For each red loop vv E R of F, join each Vi to Vi+I and v i - , . This construction is illustrated in Figure 10.5. Note that each red edge or loop of F contributes 2 to the degrees of the vertices of G corresponding to its ends, a black loop contributes 3, and a black edge that is not a loop contributes 4. It is sufficient to show that F is the frame of G, for this implies that G is BT by Theorem 10.4.3. Clearly for each vertex v of F, all the vi have the same degree in G. Therefore it only remains to show that if v and w are distinct vertices of F, then deg a v0 =fi dega w0. If v and w have the same black degree, then by Condition 2, either both have a black loop or both do not, so the black edges incident to v and w contribute equally to the degrees of v0 and w0. Also by Condition 4, v and w have different red degrees, hence deg a v0 r deg a w0 as required. Assume now that the black degree of v is larger than the black degree of w. Then by Condition 2, for each vertex x of F, z w E B implies x v E B , but not conversely. Therefore the black edges contribute more to deg a v0 than to dega w0. We now prove that for each vertex y of F, y w E R implies yv E R U B , which shows that deg C v0 > dega w0 as required. Assume that, if possible, y w E R and yv ~ R U B. Let x be a vertex of F such that x v E B and x w ~ B. Then x-~ y (sinceyv ~ B ) , hence by Condition 3, x w ~ R, i.e., x w is not an edge of F. This contradicts Condition 1. .. The construction used in the "if" part of the proof of Theorem 10.4.4 can be generalized to yield all the degree sequences of BT graphs with a given frame.
10.4
Frames of BT Graphs
269
Figure 10.5: The construction of a BT graph with a given frame.
vo . . . . ~
wo
V0
W0
Vl
Wl
-
Wl
V2 ~
W2
.,.
W2
V3 -
. W3
?-)3
.,.
vwEB
W3
vwER
V2
"
-
131
V2
"
V3
...
-
VO
Y3
.
vv E B
~
Vl
-
vv E R
VO