Homomorphism Interpolation and Approximation

Homomorphism Interpolation and Approximation

Annals of Discrete Mathematics 15 (1982) 213-227 @ North-Holland Publishing Company HOMOMORPHISM INTERPOLATION A N D APPROXIMATION Z. HEDRLiN, P. HEL...

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Annals of Discrete Mathematics 15 (1982) 213-227 @ North-Holland Publishing Company

HOMOMORPHISM INTERPOLATION A N D APPROXIMATION Z. HEDRLiN, P. HELL and C.S. KO Dedicated to Rofessor N.S. Mendelsohn on the cccasion of his 65th birthday Starting with the Interpolation Theorem for complete graph homomorphisms, we study general homomorphism interpolation. In particular, we give a set of graphs one of which is a homomorphic image of any graph with homomorphisms onto both K, and K.+l. Similar results are given for graphs with homomorphisms onto both K2 and Ch+l. We also study the notion of defect of a mapping between two graphs as a measure of nearness to being a homomorphism. We give a bound for the best defect of a mapping between two graphs, and show that deciding if there is a mapping between two graphsof defect at most k is an NP-complete problem for each k. Compatibility functions, which reflect the fine structure of defects, are introduced and briefly studied.

1. Homomorphisms

Let G and H be graphs. A homomorphism f : G - + H is a mapping V ( G )+ V ( H ) which preserves edges, i.e., such that gg’€ E ( G ) implies f(g)f(g’)€ E ( H ) . Several other definitions of the term have been proposed in the past (e.g., [5,6]), but this definition is generally accepted today as the appropriate kind of mapping between graphs. It was pioneered by Sabidussi [28,29], Hedrlin and Pultr [12, 13, 141, and Hedetniemi [ll].Homomorphisms have been studied extensively, e.g., [4, 8, 10, 12, 16, 17, 18, 19, 23, 24, 30, 311, and have been applied in such diverse areas as Ramsey theory, Ulam’s conjecture, the study of communication networks, data structures, and scheduling [3, 15, 25, 26, 27, 311. In this note we make several observations on two largely unrelated aspects of homomorphisms. In the next section we examine the amount of information on homomorphic images of a graph one can extract from the knowledge of two concrete homomorphic images. This, in a sense, extends and strengthens the Interpolation Theorem of [lo]. As a consequence we obtain some insight into the manner in which homomorphisms can increase the chromatic number. In the last section we study mappings V ( G ) + V ( H ) which are near t o being a homomorphism G + H. Such mappings are of interest if there is no true homomorphism G + H, or if it is too difficult to find one. A measure of ‘nearness’, called the defect, is investigated, and mappings with a small defect sought. We also analyze some additional parameters related to the notion of defect. 213

Z . Hedrlin. P. Hell, C.S. KO

214

Let f : G H be a homomorphism. The graph f(G) with the vertex f ( V(G)) and the edges f ( g ) f ( g ’ ) for g g ’ E E ( G ) is a subgraph of H, called the homontorphic image of G under f. If f ( G )= H, f is called an epirnorphism (a full epimorphism. [ 161). A monomorphism is a one-to-one homomorphism. The notation f : G H indicates that f is a monomorphism, and f : G + H that f is an epimorphism. The abbreviations G - , H, G - H , and G - H usually indicate t h e existence of a homomorphism, monomorphism, and epimorphism of G to H, respectively. N o t e that a homomorphism f : G -+ K , is precisely an n-coloring of G. An epimorphism f : G ++ K , is called a complete coloring [9]. (Thus an n-coloring is complete i f any two color classes are adjacent.) Evidently, if X G = t i , then every n-coloring of G is complete, and G K.. If there exists a homornorphism f : G + H, then XG S x H , because any n-coloring of H, c : H - + K,. can be composed with f to yield an n-coloring of G, c o f : G + K.. Thus G-+ H implies XG s x H . Just as colorings are sometimes described in terms of the partition into color classes they induce, so it is often useful to describe homomorphisms by their associated partitions. If f : G + H is a homomorphism, -I denotes the equiva1enc.e relation o n V ( G ) whose classes are f-’(h), h E V ( H ) . An equivalence relation o n V ( G ) whose classes are independent sets of G is called a congruence on G. Evidently, - f is a congruence o n G, because our graphs have no loops. For every congruence on G, one can define the quotient graph G/- whose vertices are the classes of and in which two classes are adjacent if there is at least one edge of G joining them. There is a canonical epimorphism f : G + G / - taking each vertex o f G to the class of containing it; clearly, is -f. Observe that a homomorphism f : G + H is a monomorphism if and only if --I is the identity relation on V(G), i.e.. every class of is a singleton. An epimorphism E : G + H for which the classes of are all singletons except for one class consisting of precisely two vertices, is called an elementary homomorphism. The existence of an elementary homomorphism G onto H is abbreviated by G A H. Intuitively, an elementary homomorphism is an identification of two (non-adjacent) vertices. Obviously. an elementary homomorphism cannot increase the chromatic number by more than one, i.e.. G A H implies xG <,yH s x G + 1 [lo]. In closing, we remark that an epimorphism between two finite graphs is always a composition of elementary homomorphisms. -+

H

-

-

-

-

-f

--F

2. Interpolation

In this section we shall often assume that G has finite homomorphic images

H I and H.. and will make deductions about further homomorphic images of G.

Homomorphism interpolation and approximation

215

First, we observe that in such circumstances G can be assumed to be finite. Indeed, suppose that HI,H,, H3 are finite and that for any finite G’, G’-W HIand G’-wHz imply G ’ + H 3 . Let G be an infinite graph, and f l : G + H l , f2 : G -W H2 be epimorphisms. The congruence on G with classes C n D, for C a class of -f, and D a class of -h, yields an epimorphism f :G G’, in which G’ = GI- is a finite graph. Moreover, G’+ HI and G‘-wHz, because each class of -f,, or -f2 is a union of classes of -. Thus G‘ + H3, and G --H H3 by composing G-w G’ and G ’ + H 3 . To illustrate the method, we prove the following well-known fact (at least, well-known for finite graphs).

-

Theorem 1 [lo]. If G + K,, and G

-W

-.,.

K,+k for some k a 2, then G -W K,,+l.

Proof. We have just observed that it is enough t o prove the theorem for finite G. Hence any epimorphism f : G + K,+k is a composition of elementary .* * 0 each of which either preserves the homomorphisms, f = E, 0 chromatic number or increases it by 1. Let GI = G, and Gi = E ~ - ~ ( G ~ - ~ ) . Since X G , +=~x ( f ( G ) ) =n + k , for some j , j = 2,. . . ,r, xGj = n + 1, and G ++ Gj + K,,+l. 0

There is an obvious aspect of interpolation to the above theorem. In particular, when k = 2, it assures a homomorphic image (Kn+l)between K,, and K,,+*.We would like to continue the interpolation, and find a homomorphic image between K,, and K,,+l-a ‘K,,+ln’. To make this precise, let n 3 2, and define t o be the class of graphs obtained from K,, by adjoining two new non-adjacent vertices a, 6, and joining a, respectively b, t o some, but not all, vertices of K,, in such a way that every vertex of K,, is adjacent to at least one of a, b (cf. Fig. 1).

Fig. 1. Some graphs in Xd+112.

Note that for any K E X,,+1,2, K + K, (identify a, respectively b, with some vertex of K,, to which it is not adjacent) and KfttK,,+I(identify a with b). Let n a 2.

Theorem 2. If G -W K,, and G -W K,,+l, then G --H K for some K E Xn+lr-. Proof. We may assume that G is finite, and f : G + is a composition of elementary homomorphisms, f = E , o E , - I ~ . . . O E ~ . Let GI = G and G, = E ~ - , ( G ~For - J .some j , xGj = n and X G , +=~n + 1. Let - c be the congruence on

Z . Hedrlin, P. Hell, C.S. KO

216

G, corresponding to some n-coloring c : G,-,K,,. We shall describe a congruence on G, with GJ- E Since G -++ G, (by . E , ) and G, --w GI/--(by the canonical epimorphism), this will prove the theorem. G,+Iis obtained from GI by identifying two vertices, say g, and g?. Since XG,,, = n + 1, any n-coloring of G,, and in particular c, must assign different colors to gI and gz. or else their common color could be kept in GI+,.Let the classes of - c be C,, C,. , . . , C,, and g, E C,,gZEC2. The congruence is defined to have the classes {g,}, {gd, C, - {gl}, C2- {g2}, C,, . . . , C,,. Any two of the classes C, - {gl}, C2- {g2}, C,, . . . , C,, are adjacent, or else G, - {gl, g2} would be (n - 1)-colorable contrary to the fact that G,,, is not n-colorable. For the same reason C, - {gl}# 8. C2 - {g2} Z 8. Thus CI- {gl), C2 - {g2Ir C,, . . . , C,, form a copy of K , in G,/-. The vertex g, is not adjacent to CI- {g,}, but is adjacent to C2- {g2} or else moving g, to C2would change c to an n-coloring of GI assigning the same color to g, and g,. Similarly, the vertex g2 is not adjacent t o C, - {gJ but is adjacent to C, - {g,}. Finally, for any i 2 3, C, is adjacent to gl or g2, or else moving both gl and g2 to C, would again change c to an n-coloring of G, in which gl and g2 have the same color. Therefore G,/- E 3y,+,i2.

-

-

Theorem 2 with n = 2 can be deduced from 1161. Theorem 2 can be viewed as a statement o n factoring homomorphisms. An n-chromatic graph G which admits an epimorphism G -,., Kn+,admits an epimorphism G K,,,, which can be factored into a X-preserving epimorphism and an elementary homomorphism. The following corollary states this more generally.

-

Corollary 3. Zf xG s n and G xH = n.

--c)

K,,,,, then G - H f , Kn+,for some H with

If XG = m 6 n, then G K , and, by Theorem 1, G --H K,,. By Theorem 2. G -,., H for some H E X,+ln; thus xH = n and H 2 K,+I.

prod.

--.I

Theorem 4. If G is a bipartite graph and G --H C Z k + l , then G --w H f, c 2 k + 1 for some bipartite graph H. We shall prove that G n K 2 and G --H c 2 k + 1 imply G + P2k+z. where Pzk+? is t h e path with 2k + 2 vertices (hence of length 2k + 1). Then clearly P2k+2$ CZir+, by identifying the endpoints. We may again assume that G is finite, and f : G --n c 2 k + 1 is a composition of elementary homomorphisms, f = F , 0 E , - , o . * .o Let GI= G and Gi = E * - , ( G ~ -Let ~ ) . j be the largest subscript such that Gj is bipartite, and let g,, gz be the vertices of Gj identified by E,. Since GI+,contains an odd cycle, g, and g2 are connected in Gi by a path P of odd length. We observe that the distance between gl and gz in Gj is at proof.

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217

least 2k + 1. Otherwise a shortest path between gl and gzin Gi, which must be of odd length because P is of odd length and Gj is bipartite, would be mapped by E~ to an odd cycle in Gj+,whose length is smaller than 2k + 1. Since a homomorphic image of an odd cycle of length < 2 k + 1 must contain such a cycle, we would obtain such a cycle in CZk+1,a contradiction. Let C be the component of Gj containing g , and g,. We have just proved that the diameter of C is d 2 2k + 1. Let go be a vertex of maximum eccentricity of C (cf. [9, p. 35]), Do= {go}, and Di, 1 S i S d, be the set of vertices of C of distance i from go. It is easy to see that each Di is an independent set in C and that Di is adjacent to Di-, and Di+,but no other Di. (Doadjacent only to D1 and Dd adjacent only to Dd-1.) Thus Do,.. . , D d are the classes of a congruence on C whose quotient CI- is P d + l . Since C+Pd+I and P,-"Pl-l (by an evident elementary homomorphism), C + P2k+*.It is easy to extend such an epimorphism t o Gi--H PZk+2 by taking a two-coloring of Gi - C and mapping all red vertices to the first vertex of Pzk+2 and all blue vertices to the second one. Finally G --H PZk+* because G --n Gj by ej-, E ~ . Note that G G', X G = xG' - 1 does not in general imply the existence of an H with X H = XG and G H InG', even when G' is critical (i.e., x(G' - g) < xG' for all g E V(G')).This can be seen by taking G' = W,, the 5-wheel, and G the graph obtained from the 5-cycle C, by adjoining 3 new vertices a, b, c and the 10 edges a l , a2, a4, b l , b3, b5, c2, c3, c4, c5.

-

0

-

0

--f*

++

Consider a finite graph G with XG = n. If f : G + Kn+l,then f = E, 0 * . . o E , and exactly one E~ increases the chromatic number. Theorem 2 asserts that the increase can be postponed till the last elementary homomorphism, i.e., that for some f' : G --n Kn+l,f' = E ~ O .. . o E ; , it is E : which increases the chromatic number. (Similar interpretation applies to Theorem 4.) If f : G ++ Kn+,, then f = E, and exactly two E ' S increase the chromatic number. However, it is no longer always possible to arrange things so that they are the last two elementary homomorphisms: In Fig. 2, it is easy t o verify that any epimor) ) 3. On the other hand, if in some f the phism E ; O E ; O E ; : G --H K4 has x ( E ; ( G = two E ' S increasing the chromatic number are consecutive, then in some other f' = E ; O 0 E ; : G ++ Kn+2 the increase occurs in E : , E ; - , . The following theorem states this more generally.

-

0

-

0

AG

€3

I

.

,

-_- , c

€2

Fig. 2.

G-K,.

E , ~ E ~ ~ E , :

2. Hedrlfn, P. Hell, C.S. KO

218

+.

Theorem 5. Let XG = n and assume that f : G K,,, is the composition f = fu 0 F, 0 F , - , 0 . . . o fa where each E,, i = 1, . . . , r, is a n elementary homomorphism increasing the chromatic number. Then there is a n f‘ : G-w K,,, f ’ = C:OE;. . . O E ’ , o f m where each E : , i = 1, . . . , r, is a n elementary homomorphism iwreectsing the chromatic number. 0

0 .

Proof. Note that fu, fw, fb preserve the chromatic number. Also note that Corollary 3 implies Theorem 5 for r = 1. We may assume that f = fw E, 0 . . * o el, or else we could replace G by f,(G). Let GI = G and G, = E,.,(G,-,); let the vertices of GI identified by E , be g, and g:. Since e = F , O . . ’ O E , increases the chromatic number of G by r, there are at least r vertices u in e ( G ) = G,,, with le-’(L>)l> 1. This implies that E ] , . . . , E , identify disjoint pairs of vertices of G, i.e., that gl, . . . . g,, g ; . . . . , g : are distinct. Let S = {gl,. . . ,g,, g ; , . . . , g : } . Let CI,Cz,. . . C,, be the classes of some n-coloring G 4 K,,. Consider the congruence on G whose classes are {gl}, . . . ,{g,}, {gi}, . . . ,{g;}, CIS, . . . , C,, - S. Let G’ = G/- and f : : G --n G’ the canonical epimorphism. Clearly, xG’ = n because each g,, respectively g : , is not adjacent to some C, - S. (Namely for j such that C, contains g,, respectively g : . )Let E : identify g, and g: and let f = E : O . . . o E ; o f : ; we shall show that f’(G)= K,,,. Since xG,, = n + r. any (n + r)-coloring of e( G ) = G,,I is complete. Let g’i be the vertex of e ( G ) resulting from t h e identification of g, and g: by E l . Then {g:}. . . . { g ; } , CI- S, C2- S, . . . , C,,- S are the classes of a complete (n + r)coloring of e(G), and consequently { g , , gi}. . . . , (g, g:}, Cl - S. C, - S, . . . , C,, S are the classes of a complete (n + r)-coloring of G. On the other hand, they are precisely the classes of -,.. Thusf’(G) = K,,,. Consequently, each E : increases the chromatic number by 1. 0

-

.

.

-

A strongly complete n-coloring of G is a coloring c : G K, in which every color class contains a vertex adjacent t o all other color classes. If XG = n, it is easy to see that every n-coloring of G is strongly complete. Conversely, if G admits complete n-colorings and each complete n-coloring is strongly complete. then XG = n by Theorem 2. (Every K E X(n-l)+l,2 has a complete n-coloring which is not strongly complete; if G K. then G also admits such a coloring.) This observation is generalized to multicolorings in [21]. --H

3. Approximation

In this section we shall consider only finite graphs. In practical applications, a homomorphism f : G + H may correspond t o a desired arrangement of objects, or schedule of activities. The general problem of existence of a homomorphism f : G + H is NP-complete, because it includes the question of

Homomorphism inferpolation and approximation

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-

n-colorability of G (taking H = K,,) [20]. Of course, in special cases the existence of a homomorphism f : G + H is easy to determine. For example, among bipartite graphs always G H, because G + K 2 - H. For trees, there are polynomial time algorithms to determine whether T ++ T’ [22]. Still, even for trees, it is an NP-complete problem to determine whether T + T‘ [7]. If there does not exist a homomorphism f : G -+ H, or if it requires too much effort to find one, we may wish to find a mapping f : V(G)-+ V ( H ) which is near to being a homomorphism. One way to measure such nearness is the number of edges to be removed from G and/or added to H to make f into a homomorphism, f : G --* H. To be more specific, let G and H be graphs, and let

G’ = {G’1 V(G’) = V ( G ) ,E(G’) E ( G ) , (E(G)- E(G’)I S i } ,

H, = {H’ I V ( H ’ )= V ( H ) ,E ( H ’ )2 E ( H ) , IE(H’)- E(H)I s j ) . The set Homj(G, H) consists of all homomorphisms f : G‘+ H’ for all G‘ E G’ and H E 4.Thus Homt(G, H) is the set of all homomorphisms of G to H, and f E Homj(G, H ) if and only if one can remove some S i edges of G and add some S j edges to H in such a way that f becomes a homomorphism. Let f : V ( G ) + V ( H ) be any mapping. We define the negative defect, d - ( f ) = min{i 1 f E Homb(G, H ) } , the positive defect, d+cf)= minG I f E Hom:(G, H)}, and the defect, dcf) = min{i + j I f E Homf(G, H ) } . Note that when discussing the various notions of defect for a mapping f , it is important to know not only V(G) and V ( H ) ,but also G and H ; for instance any f : V ( G ) + V ( H )is of defect 0 as a mapping from the graph on V ( G )which has no edges. Therefore we shall always discuss the defects of a mapping between two graphs. In order not to confuse such mappings with homomorphisms, we shall use the notation f : G -+ H to indicate that f is a mapping of the vertex-set of the graph G to the vertex-set of the graph H. The notion of negative defect was used in an inclusion-exclusion argument by Muller to verify the edge-reconstruction conjecture for a large class of graphs [25]. From now on, we shall always assume that G and H have the same number of vertices, denoted by p.

Theorem 6. Let xG = n, XH = m, and p = 9m + r for non-negative integers 9 and r, 0 =Z r < m. (a) The minimum positive defect of any f : G-+ Kpis (1). (b) If K,,, is a subgraph of H, then the minimum negative defect of any f :K,-+Hism(j)+qr. Proof. (a) The minimum positive defect of any f : G -+ Kp is the minimum j such that G + G‘ for some G‘ with p vertices and j edges. Such a G‘ is then a

2. Hedrlin. P. Hell, C.S. KO

220

‘minimum range’ of G. Since xG = n, G - , K,,, and K,, has (1) edges. Moreover, if G 4 G‘, then x G ’ n~and hence V ( G ’ )can be partitioned into at least n classes with at least one edge between any two classes. Thus G‘ has at least (?) edges. (In fact, it is not possible that G’ has precisely (1) edges unless G‘ = K , ; hence Kn is the only minimum range of G. The analysis of minimurn ranges is more interesting for digraphs [18].) (b) The minimum negative defect of any f : K p+ H is the minimum i such that H ’ - H for some H’ with p vertices and (5)- i edges, i.e., (5) minus the maximum j such that H ‘ + H for some H’ with p vertices and j edges. Such an H‘ is then a ‘maximum domain’ of H. Since K,,, is a subgraph of H, H’+ H if s rn. Thus the maximum domain H’of H will be the largest and only if xH’ m-partite graph with p vertices. Hence H’ is the complete m-partite graph with all parts as equal as possible. If p = 9m + r, 0 G r < m, then r parts will have 9 + 1 vertices and m - r parts 9 vertices. Therefore, the minimum negative defect is the number of edges in H ’ , which is r(q;‘)+ ( m - r)(i) = m(;)+ 91. Since adding edges t o H and removing edges from G cannot increase any kind of defect, we have the following corollary.

Corollary 7 . Under the circumstances of Theorem 6, there always exists a mapping f : G--*H of positive defect at most (1) and a mapping f’: G - +H of negative defect at most m(f) + 91. Both bounds are bestpossible, i.e.. are achieved for some pairs G, H. Let No= (0. I , ? , . . .} and G, H be fixed graphs with p vertices. We shall define a function CG.”: No+No which captures the tradeoff, in finding a homomorphism of G to H, between the deletion of edges from G and the addition of edges to H: For any j E No,

C , , ( j )= min{i 1 Homj(G, H) # 8) Thus C , & ) is the smallest number of edges whose deletion from G allows a homomorphism to H with some j edges added, i.e., C G , , ( ~ ) = min{d-Cf) I f : G -+ H’ with H‘ E H,}. Also note that min{d(f) I f : G - +H} = minG + C,,,(j) I j E No}. Clearly, each C,,is a decreasing function No-, No.

Lemma 8. Let 1

In particular, if p

m

=

dp

qm,

and p

= qm

+ r, 0 < r < m. If

(7) s j < (“‘;I),

then

Homomorphism interpolation and approximation

22 1

Proof. Let (7) s j < ('"Tl).As observed above, we seek the minimum negative defect d - ( f ) of any f : Kp-+H with IE(H)(S j . Since such an H has fewer than ('"$I) edges, it is m -partite. Hence, any G with G + H must also be m -partite and therefore has at most m($)+qr edges. Thus d - ( f ) 3 m ( f ) +qr for any f : Kp-+ H. Let H be the disjoint union of K , and Kp-,,,.According to Theorem 6(b) there is a mapping f : Kp-+ H of negative defect m (5) + qr. This proves the lemma. (We note that one can prove that, for in 2 5, a graph H with fewer than ('"TI) edges which does not contain K , is (m - 1)-colorable. Indeed, if X H = m, then any m-critical subgraph of H would have at least m + 2 vertices [2, p. 1211 and at least i((m - l)(m + 2) + ( m- 3))3 ('"l') edges [ 1, p. 2361. Thus the smallest negative defect can only be attained by placing the j edges so as to form the largest possible complete subgraph in H.)

Theorem 9. ntere always exists a mapping f : G-+ H of defect at most

and the bound is asymptotically best possible, because the minimum defect of a mapping f : Kp-+ k7, is greater than 3(32)-'/3p4"- ;p. Proof. We computed CKP%in Lemma 8. From there we see that I j E NO}occurs when j = (7) for some min{dCf) I f : Kp-+Ep}= minG + CKPzp(j) m = 1, . . . ,p. Thus we need to minimize the function

where p = mq + r, 0 s r < m, and 1s m s p . Since q = I p / m ] , (p - m ) / m < q , s p/m, and

Therefore,

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& I . .l

Z. Hedrlin. P.Hell, C.S. KO

The minimum of Am2+{p’/m occurs at m = 2-1/3p2/3, and is 3(32)-’/3p4/9.Hence the minimum of f ( m ) is strictly between 3(32)-”3p4’3-zp and 3(32)-”3p4/3f $p. It is clear that the minimum defect of any f’ : G -+ H is at most the minimum defect of a mapping f : K,, -+ K,. We have found that on the one hand, it is difficult to determine if there is a mapping f : Ci--.H of defect 0 (i.e.. a homomorphism f : G + H), and on the other hand, there always is a mapping f : G - + H of defect at most 3(32)-”3p”3+;p. It is conceivable that for some large fixed k, the problem of existence of a mapping f : G -+ H of defect k becomes easy. That, however, is not the case. For k E No, we denote by DEF(k) the decision problem: Given G and H (with the same number of vertices), does there exist a mapping f : ( 3 - + H with d ( f ) c k ?

Theorem 10. DEF(k) is NP-complete for any k E No. Before we embark on a proof. we introduce some concepts that shall also be used later. Let S be a graph, t 3 1 a n integer. A sequence of vertices xo, xl,. . . , x, of S is a power-t-path from xo to x, in S. if z 2 t, and ax, E E ( S ) whenever 0 < Ii -jlc t. A graph S is power-t-connected [17] if any two vertices of S are joined by some power-t-path in S. We observe that if S is power-t-connected, r 2 2. then for any edge e of S, S - e is power-([ - 1)-connected: Consider two arbitrary vertices x and y of S and a power-t-path x = xg, x,. . . . , x, = y joining them. The same sequence of vertices is a power-(t- 1)-path in S - e, unless e = x,xb with a < h and a f 0 or 6 f 2. Since the situation is symmetric, we assume that a # 0. A power-(1 - 1)-path joining x and y in S - e can be obtained by removing x, from x,), xi, . . . , x,. We also note that the homomorphic image of a power-r-connected graph is likewise power-t-connected. Finally, observe that a power-t-connected graph contains K,,,. Let S, be the graph with vertices 0, 1, . . . , 3 t + 3 and edges ij, for li - jl S f, and 0. 3t + 3. It is easy to see that ,yS, = t + 2 and, in fact, S, is ( t + 2)-critical. Also. S, is power-t-connected, and hence S, - {el, . . . , e,} is power-(t - s)connected, for any set of .F edges { el , . . . ,e,} of S,.

Proof of Theorem 10. Evidently, DEF(k) E NP. Since DEF(0) is the problem of existence of a homomorphism G + H, it is NP-complete. (The fact that G and H have the same number of vertices poses n o problem, since G-+ K, U&, if and only if G-, K,. which we have earlier observed to be NP-complete.) We shall exhibit a polynomial reduction of DEF(k) to DEF(k + 1); then each DEF(k) is NP-complete by induction. Given graphs G and H with the same number o f vertices, we shall construct, in polynomial time, graphs G’ and H’

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Homomorphism interpolation and approximation

with the same number of vertices, such that there is a mapping f : G - +H with defect d ( f )=sk if and only if there is a mapping f’ : G ’ - + H ’ with defect d ( f ‘ ) s k + 1. Let I V(G)( = I V(H)l=p and t = p + 2k + 3. Let G’ be obtained from G by attaching a copy of S,, by vertex 0, at every vertex of G, and adding to one copy of S, the additional edge t + 2, 3t + 3. Let H’ be obtained from H by attaching a copy of S , by vertex 0, at every vertex of H (cf. Fig. 3). Any mapping f : G - + H of defect dCf)s k can be extended to a mapping f’ : G I - + H ’ of defect =sk + 1, by mapping the vertex i of the copy of S, attached at g E V ( G )to the vertex i of the copy of S, attached at f(g) E V(H). Conversely, consider a mapping f’ : G’-.+H’ of defect d(f‘) S k + 1. Let f’ E Homj(G’, H’) with i + j = d(f‘). Thus f’ is a homomorphism f ’ : G”+ H” where G” is obtained from G’ by deleting some i edges and H” from H’ by adding some j edges. Let V be the set of vertices of any one copy of S, in G’. Since S, is power-t-connected, and t - i 3 t - k - 1 = p + k + 2, V induces in G“ a power-@ + k + 2)connected subgraph. Therefore f’(V) also induces a power(p + k + 2)-connected subgraph of H”. Note that H’ is obtained from H” by deleting j S k + 1 edges-thus f’(V) induces in H’ a power-@ + 1)-connected subgraph. It follows that f’(V) is a subset of some copy of S, in H’. Hence we can define a mapping f : G - + H by setting f(g), g E V ( G ) ,equal to h if the copy of S, at g is mapped by f’ into the copy of S, at h. It remains to show that d(f)=sk. Note that every edge of G’ is either an edge of G or an edge of a copy of S,. Of the i edges removed from G’, let il be in the copies of S, and i2 in G ; of the j edges added to H’ let jl be added to the copies of S, and j z between the copies. (Note that this latter category includes edges added to H.) Thus i, i2 = i and jl + j 2 = j. Then il + j l 2 1 because there is no homomorphism of the special copy of S, with the edge t + 2, 3t + 3 t o S,.(There is obviously n o monomorphism, and xS, = t + 2 while X S z s t -t 1for a subgraph S of S, which does not include all vertices of S,.)Thus d ( f )S i2 + j 2 = (i + j ) - (il + j , ) S d ( f ’ )- 1s k .

+

5

5

G’

H‘ Fig. 3. An illustration of G , H’ with t = 2 (in actual constructions I = p + 2k

+ 3).

Z. Hedrlin. P.Hell, C.S. KO

224

Our final observations concern the functions C,, It is obvious from the definition, that each function C G ~ H: No+No is decreasing. Of course, it is not strictly decreasing. in fact there is an s E Nosuch that C G , H ( t ) = 0 for all t > s. The unique smallest s with this property is called the support of C , H . We shall prove that every decreasing function c : No-+Nowith finite support is equal to CG,Hfor some graphs G, H . The first step of the proof consists of proving the assertion for digraphs. Homomorphisms of digraphs are defined in the expected manner, as mappings of vertices which preserve t h e arcs; G' and Hj are defined as for graphs, except that we count the arcs added/deleted, rather than the edges. The definitions of Homj(G, H ) and C , , then apply directly.

Theorem 11. Each decreasing function c : No-+No with finire supporr is equal C& for some digraphs G. H .

To

Proof. Let 7' be the digraph with vertices 1 , 2 , . . . , k, and arcs ij for i < j . (Thus Tk is the transitive tournament of order k.) For i = 0, 1 , . . . , k - 1, let T i

be obtained from Tk by the addition of the vertex 0 and the i arcs . . , Oi. Thus TO, is Tk with the isolated vertex 0. We claim that the function C,-;,T;(j)has value 1 for j s i - 1 and value 0 for j i. Indeed, T i - 01 + TO, (by identifying 0 and l), thus C T ; , T i ( j ) S 1 for all j . Moreover C T ; , T i ( i ) = 0, because T i with the additional edges 01,. . . , O i allows a homomorphism (in fact, isomorphism) from TI. It remains to show that if T is obtained from Tk by the addition of at most i - 1 edges, then T i tr T. This follows easily from the observation that T i does not contain two arcs ab, a f b f with none of the arcs aa', a'a, bb', and b'b. (No two arcscan then be identified by a homomorphism and T i + T implies that T i has at least as many arcs as T.)

01,0?,.

Let c : hi,,-+ Nobe a decreasing function of support s. We may assume that c is not identically zero, or else CT,,J, = c. Thus s 2 1; let k = s + 1. We define

G

7

~ ( -k2)Ti-I U(c(k

- 3 ) - c ( k - 2))T:-*U *

*

U(c(0) - c(1))T:

where U is the disjoint union and xT = T U T U . UT, x times. Since c(0) # 0, not all coefficients in the expression for G can simultaneously be zero. Let H be the disjoint union of T! and (c(0)- l)(k + 1) isolated vertices; evidently, 1 V(G)I = 1 V(H)I. The value of CG,,(j)is the minimum number of arcs that need to be removed from G to allow a homomorphism to some H' obtained from H by the addition of some j arcs. It is easy to deduce from our calculations of C T L that ~,

Homomorphism interpolation and approximation

225

if j s k - 2, and CG,,(~) = 0 = c ( j ) for j 3 k - 1 = s. (A crucial observation for the former conclusion is the fact that the same j edges u l , u 2 , . . . , uj added to H allow a homomorphism to H from each of T i , T:, . . . , T i . ) One may require G and H to be connected (cf. Fig. 4): It suffices to add to G one new vertex w and arcs wk to all vertices k of the copies of T i , and to H one new vertex z adjacent to k and all isolated vertices. It is easy to see that this transformation does not affect C,,.

A+

z

w

1 3+ 4 5 . 3

0

0

1

0

G

Fig. 4. Connected digraphs G, H, yielding the function c defined by c(O)= 3, c ( l ) = 2, c ( j ) = 0 for j 3 2.

Corollary 12. Each decreasing function c : No+ No of finite support is equal to CG,H for some (conneckd) graphs G, H.

Proof. Let G', H be digraphs with C G , , H '= c, and let s be the support of c. We may again assume that s 1. Let t = 2s + c(0) and let G be obtained from G' by replacing each vertex by a copy of Sr and replacing each arc xy by an edge joining the vertex t + 2 of the copy of S, which replaces x, with the vertex 3t + 3 of the copy of S, which replaces y. Let H be obtained from H by the same construction. We claim that C , , = CG',w (= c). (Note that since G', H' have the same number of vertices, so do G and H. Moreover, if G' and H' are connected, then so are G and H.) Clearly, for any mapping f E Homj(G', H'), there is a corresponding mapping f € Homj(G, H), taking the vertex i of the copy of S, which replaces x to the vertex i of the copy of S, which replaces f ( x ) . Thus C , H ( j )s Cc,w(j) and CGvH(j) = 0 (= CG,H,(j))for j 2 s. On the other hand, for j < s, the addition of any j edges to H results in an H in which only subsets of the copies of Sr are

2x7

Z . Hedrlin. P. Hell. C.S. K O

power-2s-connected. (Any set of vertices which is power-2s-connected in I? is j ) < CG..Hs( j ) = c(j ) s power-(s + 1 )-connected in H . ) If. for some j < s. CG,H( c(0) then some G obtained from G by the deletion of fewer than c(0) edges would allow a homomorphism into some fi. Since in any such G, the copies of S, remain power-2s-connected, they would be mapped i n t o copies of S,. It is now easy to mimic the situation in G' and H'-adding some j edges to H' and deleting some C c ; , H ( j < ) C GH . ( j ) e d g efrom ~ G' resulting in a homomorphism, contrary to the definition of C ( ; . . H . .Hence CG,H = CG..H. = C. Charles University Prague. Czechoslovakia Simon Fraser University Burnaby Vancouver, Canada Rutgers University Newark. NJ. USA

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