MATHEMATICAL

COMPUTER MODELLING

PERGAMON Mathematical and Computer Modelling 32 (2000) 1071-1082 www.eIsevier.nl/locate/mcm

Sphere-Of-Influence Graphs Using the Sup-Norm

E. BOYER* Lyon College

Batesville, AR 72501, U.S.A.

L. LISTER Bloomsburg University of Pennsylvania

Bloomsburg, PA 17816, U.S.A.

B. SHADER University of Wyoming

Laramie, WY 82071, U.S.A.

(Received March ZUOO; accepted April 2000)

Abstract-The SIG-dimension of sphere-of-influence graphs using the sup-norm is investigated. Constructions which give an upper bound on the SIG-dimension of arbitrary graphs are given. In addition, lower bounds, which are often tight, on the SIG-dimension of complete multipartite graphs are derived. @ 2000 Elsevier Science Ltd. All rights reserved.

Keywords-Sphere-of-influence, Graphs, Sup-metrix.

1. INTRODUCTION

The notion of sphere-of-influence graphs was introduced by Toussaint in the area of computer

vision and pattern recognition [l]. A sphere-of-influence graph is defined to be a set of points,

each with an open ball centered about it of radius equal to the distance between that point and

its nearest neighbor. Two points in the graph are adjacent if their open balls intersect. One very

basic question in this area is the following: given an arbitrary graph G, does there exist a set of

points in the plane so that G is isomorphic to the sphere-of-influence graph determined by this

set of points? The answer to this question is unknown and appears to be quite difficult (see [2]).

In the original setting, points are in the plane and distance is determined using the Euclidean

metric. As noted in [3], it is natural to extend these ideas to higher dimensions and to other

metrics. Some of these extensions have been studied, see for example [4-71. It is interesting to

note that changing the metric significantly changes the problem.

In this paper, the metric induced from the sup-norm is considered. It is known [5] that if G

has no isolated vertices, then G is isomorphic to a sphere-of-influence graph using this metric for

some dimension d. We examine how large d must be for an arbitrary graph of order n which has no isolated vertices.

*Partially supported by a Christian A. Johnson Fellowship.

0895-7177/00/g - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. Typeset by 4&W PII: SO895-7177(00)00191-6

1072 E. BOYER et al.

Throughout, we use the notation and basic definitions related to graphs in [8].

Let p be the metric in II@ given by

p(z, y) = max {I

z(i) - yci) : i = 1,2,;.. , d , 1

where z = (x(~),x(~), . . . , dd)) E Rd and y = (y(l), ~(~1,. . . , ~(“1) E Rd. Consider a set V =

{% 212, . . . , w,} of n points in Rd. For each i, define ri as

~i=min{p(wi,~+):j=1,2 ,..., n;j$i}.

The ball of radius ri about Q, BTi(ui), is the set of all points z E Wd such that p(vi, Z) < ri. The

sphere of influence graph, M&SIG(V), is the graph with vertex set V and edge set

{Viuj : B,i(G)r)&j (vj) # 0; i #j} *

Equivalently, vivj is an edge in M&SIG(V) if and only if i # j and p(vd, uj) < ri + Tj.

A graph G is an M&SIG graph if there exists a set of points V = {q, ~2,. . . , v,} 2 I@ such

that G is isomorphic to M&SIG(V). In this case, we say that G is realized by V and that G is

realizable in Rd. If G is realized by V in I@, then for each positive integer e the graph G. is realized

by V’ in Rd+e, where V’ is the set of points obtained by appending e zero coordinates to each

point in V. Clearly, a necessary condition for a graph with at least two vertices ti be realizable

is that it does not contain isolated vertices. Furthermore, each graph G with no isolated vertices

may be realized in II@ for some d since the rows of the matrix 2I+ A realize G, where A is the

adjacency matrix for G and I is the identity matrix. Thus, for each graph G with no isolated

vertices,

SIG(G) = min {d : G is realizable in I@}

is well defined, and is called the SIG-dimension of G. Indeed, as noted in [5], the rows of the

matrix obtained from A + 21 by deleting its last column are a realization of G in,,lP-‘, where n

is the number of vertices of G. In other words,

SIG(G) 5 n - 1. (1)

Note that if G is a disconnected graph with connected components G1, G2, . . . , GI, and m =

max{SIG(Gl), SIG(G2), . . . , SIG(Gk)}, then by realizing each Gi in B” by point sets that are

sufficiently far apart, G can be realized in Rm, and hence, it follows that SIG(G) = m.

In this paper, we study the SIG-dimension of graphs. In Section 2, tie show how a certain type

of edge covering, a distinguished biclique cover, can be used to give a realization of a graph. As

a consequence, we improve the upper bound given in (1).

In Section 3, we show how to construct a realization of a graph G from a realization of a

subgraph of G obtained by deleting a vertex. We show that adding a vertex to a graph increases

the SIG-dimension by at most one. This construction is generalized to one that realizes a graph G

from a realization of one of its realizable induced subgraphs.

In Section 4, we establish lower bounds for, and in some cases determine, the SIG-dimensions

of complete multipartite graphs.

The results from these sections indicate that there may exist a constant k < 1 such that the

SIG-dimension of a graph of order n is bounded above by kn. In particular, we make the following

conjecture.

CONJECTURE. Let G be a graph of order n with 110 isolated vertices. Then

SIG(G) 5 f . 1 1 In Section 5, we discuss this conjecture and give numerous families of graphs which satisfy this

conjectured bound.

Sphere-of-Influence Graphs 1073

2. DISTINGUISHED BICLIQUE COVERS

In this section, we introduce a new type of graph covering, and use it to construct realizations

of a graph.

A biclique, B(R, S), of a graph G is a subgraph of G which is a complete bipartite graph with

bipartition (R, 5’). A bicliqve cover of G is a collection

B (RI, Si) , B (R2, s2) , . . . , B (Rd, sd)

of bicliques of G such that each edge of G is in at least one biclique. We define a distinguishing bicliqve cover to be a biclique cover of G such that for each edge xy of ??, there exists at least

one biclique B(Rk, Sk) such that

In other words, for each pair of nonadjacent vertices of G, there is at least one biclique that

distinguishes the two vertices in the sense that one of the vertices is a vertex of the biclique and

the other is not. In this setting, we allow one of the sets R or S to be empty. The biclique cover number, be(G), is the least number of bicliques needed to cover all the edges of G. The

distinguishing biclique cover number, be*(G), is defined similarly. Clearly, be(G) 2 be*(G).

Suppose that G has vertex set {1,2,. . . , n} and that

B(Rl,Sl),B(Rz,Sz),...,B(Rd,Sd) (2)

is distinguishing biclique cover of ??. A realization of G in lRd can be constructed as follows. For

i = 1,2,. (1) (2) ..,nletvi=(vi ,wi ,..., , v!~‘) be the vector in lRd where

{

0, if i E Rk,

?J?) = z 1, ifi$!&USk,

2, if i E Sk.

Note that each entry of 2ri is a 0, 1, or 2.

The definition of the vis and the fact that (2) is a biclique cover of G implies that if ij is not

an edge of G, then p(vi, vj) = 2. The fact that (2) distinguishes nonadjacent vertices of E implies

that p(vi, vj) = 1 whenever ij is an edge of G. Thus, since G has no isolated vertices, ri = 1 for

all i and it readily follows that V realizes G.

We note that if G has vertices 1,2, . . . , n, then the stars of c at vertices 1,2,. . . , n - 1 form

a distinguishing biclique cover of ??, and the vectors ~(~1 are precisely the rows of the matrix

obtained from A + 21 by deleting its last column, where A is the adjacency matrix of G. Hence,

the construction in [5] used to prove (1) is a special case of the above construction. As a further

consequence, we have the following theorem.

THEOREM 1. Let G be a graph of order n with no isolated vertices. Then

SIG(G) 5 bc* (q .

We now use this theorem to study the SIG-dimension of complete graphs, and complete mul-

tipartite graphs. Note that distinguishing biclique covers of K are related to biclique covers

of K,. Namely, if (Xi, Yr), . . . , (xk, Yk) is a biclique cover of K,, then (Xi, 0), . . . , (xk, 0) is a

distinguishing biclique cover of G, and if (Xi, a), . . . , (xk, 8) is a distinguishing biclique cover

ofK, then (Xi,%),.. . , (xk, z) is a biclique cover of K,. It is known that bc(&) = [log, n]

(see [9]). Thus, there exists a distinguishing biclique cover of K of the form

(X1,0),(X2r0),...,(Xrloganl,0), (3)

and by Theorem 1,

SIGK,) 5 Pog,nl . (4)

In Section 4, we show that equality holds in (4). For complete multipartite graphs, we have

the following upper bound.

1074 E. BOYER et al.

COROLLARY 2. Let nl,nz,. . . ,ns 2 2 be integers, and let no ml =..e=rn,, =l. Then

be an nonnegative integer. Set

if no = 0, and,

if no > 0.

PROOF. Let G = Knlrn2 ,..., n,,ml,m2 ,..., m,,, . If G is a complete graph, then the result follows

from (4).

Assume that G is not a complete graph. The complement of G is the disjoint union of the

complete graphs K,, , K,,, . . . , K,* and no isolated vertices. The graph K,; can be covered with

a set Si of [log, nil bicliques. If no > 0, then by (3) there is a distinguished biclique cover of

the no isolated vertices consisting of a set, SO, of [log, no1 bicliques. ,The result now follows

from Theorem 1 by verifying that the set of bicliques in U&,$ is a distinguishing biclique cover

of G. I

In Section 4, the upper bound in Corollary 2 is shown to be tight if ~$0 5 1. The next result

shows equality need not hold in general.

PROPOSITION 3. Let k, s, and n be positive integers with s < 2” + 1, and 11 5 2”. Then the

SIG-dimension of the complete multipartite graph with s partite sets of size 1 and one set of

sizenisatmostk+l.

PROOF. Consider the n+s by k+ 1 matrix h/l where rows 1 through n are n distinct (0,2)-vectors

with the last entry equal to zero, row n + 1 is Ic ones followed by a zero and rows n f 2 through

n+ s are s - 1 distinct (0,2)-vectors with last entry equal to 2. We consider the SIG-graph defined

by the rows of M. Any two of the first n rows are at distance 2 apart, as are any two of the last s

rows. Any of the first n rows and any of the last s - 1 rows are at distance 2 apart, And any of

the first n rows are at distance 1 from the (n + l)st row. It follows that a vertex corresponding

to one of the first n + 1 rows has radius 1, and to one of the okher rows has radius 2. Hence, the

rows of M realize the complete multipartite graph with s partite sets of size 1 and one partite

set of size 71. I

We now derive upper bounds for the SIG-dimension of graphs in terms of the size of a largest

clique. Note that in checking that a biclique cover of a graph G is a distinguishing biclique cover,

we must check that each pair of vertices i and j which are adjacent in G are distinguished by

some biclique of the cover. Certain conditions allow one to eliminate some of this checking. For

example, if one of the bicliques is the star at i, then each pair of adjacent vertices in G containing

i is distinguished.

COROLLARY 4. Let G be a graph of order n with no isolated vertices, and let H be a clique of G

of order t. Then

SIG(G) 5 n - t + [log, tl .

PROOF. Let {1,2,. . . , n} be the vertices of G and suppose that {n-t + 1,. . . , n} are the vertices

of H. By (3), there exists a distinguished biclique cover

(Xl 7 0) I (X2,0) 7 . . ., (qog* t1 I0) (5)

of z. The result follows from Theorem 1 by verifying that the biclique cover of ?? which consists

the bicliques in (5) and the stars centered at {1,2,. . . , n - t} is distinguished. I

Corollary 4 can be improved if each pair of vertices of H have different sets of neighbors. More

precisely, let G be a graph with vertex set V and edge set E. The closed neighborhood of a

vertex 21 of G is the set N[v] = {U E V : uv E E} u {v}.

Sphere-of-Influence Graphs 1075

If i and j are adjacent vertices of G whose closed neighborhoods are different, then each biclique cover of G will contain a biclique which distinguishes i and j.

COROLLARY 5. Let G be a graph of order n with no isolated vertices. Let H be a clique of G of order t such that no two vertices of H have the same closed neighborhood in G. Then

SIG(G) < n - t.

PROOF. Let {1,2,. . . , n} be the vertices of G and suppose that {n - t + 1,. . . , n} are the vertices of H. Consider the biclique cover of 21’ which consists of the stars centered at {1,2,. . . , n - t}.

Every edge of G either contains at least one vertex which is the center of one of the stars in the biclique cover or joins two vertices which do not have the same closed neighborhood. Hence, this biclique cover is a distinguishing biclique cover of c with n - t bicliques. The result follows from Theorem 1. I

3. REALIZATIONS VIA INDUCED SUBGRAPHS

We now describe a construction that is based on realizations of induced subgraphs. For v E Rd and a E R, we let [v, a] denote the vector in lR d+l whose first d coordinates are those of v, and whose last coordinate is a. The following lemma gives a family of realizations of a graph in Rd+r from one realization in Rd.

LEMMA 6. Let G be a graph of order n with no isolated vertices. Suppose that G is realized by V = {v~,vz,. . . , v,} C Wd with radii 7-1,7-z,. . . , r,. Let R = max{ri 1 i = 1,2,. . . , n}. For i = 1,2,..., n, let ai be a number such that R - ri < ai 5 R and let wi = [vi, ai]. Then G is realized byW={wl,...,~~} inRd+‘.

PROOF. First observe that ai -aj < R-(R-TO = Ti 5 p(vi, vj), and similarly, aj -ai _< p(vi, vj). Hence, it follows that P(w~, wj) = p( v v. i, 3) f or all i and j. Therefore, vivj is an edge in M&SIG(V) if and only if wiwj is an edge in iWe1 SIG( W) . I

The next theorem bounds the SIG-dimension of a graph in terms of the SIG-dimensions of certain induced subgraphs.

THEOREM 7. Let G be a graph with vertex set V and which has no isoltlted vertices. Let x be a vertex of G, and let H be the subgraph induced by V \ {x}. Let HI,. . . , Ht be the connected components of H of order 2 or more and Jet p be the number of isolated vertices of H or one, whichever is greater. Then

SIC(G) I I + max {[log, (p)l , SIG (HI) , SIG (Hz), . . . , SIG (Ht)} .

PROOF. If t = 0, then G E Ki,,. By Corollary 2, SIG(Ki,,) I [log,pl and the theorem holds. Now assume that t 2 1.

Let d = max{ [log, (p)l, SIG(Hr), SIG(Hz), . . . , SIG(Ht)}. For i = 1,2,. . . , t, let ni be the

number of vertices in Hi and let V(i) be a set of ni vectors in Rd which realizes Hi. Since a set of vectors which realizes a graph may be scaled or translated, we may assume that each entry in each vector in V(i) is between 3i and 3i + 1.

The graph realized by v = UV(i) for i = 1,2,. . . , t is the graph consisting of the disjoint components HI, Hz, . . . , Ht. Let G, be the induced subgraph of G consisting of HI, Hz, . . . , Ht and vertex x. Suppose that {1,2,. . . , t, l+ 1) are the vertices of G,, vertex x is vertex e + 1, and without loss of generality, suppose that { 1,2,. . . , lc} C { 1,2, . . . ,l} is the set of vertices adjacent

to x. For i = 1,2, . . . , l, let iii be the vector in v corresponding to vertex i and let fi be the

radius associated with 6i in M&(c). Set R = ma.xl<i<e{Fi}, and for i = 1,2,. . . , I, define

1076 E. BOYER et al

Next set M = max{R, p(cl, 5i) : i = 2,. . . , k} and ve+l = [cl, R + M]. Let V = {wl, v2,, . . , q}. We claim that Mz’(V U {we+l}) is isomorphic to G,.

Let ri denote the radius of vertex vi in ii& ‘+‘(VU{ve+l}). Since ~(ve+~, wi) = lllax(p(6~ ) ai), Al}

= M for 1 < i 5 k and ~(ve+l, Q) = max{p(6l,&), M + ri} for k + 1 < $ f C, re+l = M. Given

a vertex i, with 1 5 i 5 e, there exists a vertex j with 1 < j 5 f? such that p(ci, cj) = pi. Thus,

for this j, p(wi, wj) = r”i and for any m # j, p(wi, v,) 2 r’i. Therefore, ri = fi for i = l,2,. . . , n.

It follows as in the proof of Lemma 6 that for i,j 5 e, i and j are adjacent in M&t-l(V u {vel})

if and only if i and j are adjacent in M&(p).

For 1 5 i < Ic, ~(ve+l, vi) = max{p(fil,6i), M} = M < re+l + ri. Hence, vp+lzli is an edge in

M$+“(V U {ve,}). Also, for k + 1 5 j 5 e, ~(ve+l, wj) > M + Fj = re+l + rj. Thus, ue+lvj is not

an edge in M$‘(V U {we+l}). Therefore, Mm ‘+‘(V U {ve+~}) is isomorphic to G,.

We now consider the isolated vertices of H. If H has isolated vertices, let d = (61,. . . , iip} be a

set of distinct (M, -M) vectors in R ‘+I whose (d + 1)st entry equals M. Since 8 u { (0, 0, . . . ,O)}

realizes Kl,, in R d+l, U U {vg+l} also realizes Kl,, whereU=U+ve+l={iit+l+~~+l,...,tip+

we+l}. If H has no isolated vertices, let U be the empty set. We claim that U U V u {~e+~}

realizes G in @+I.

First, note that the radius associated with ve+l in the ‘set U U V U {ve+l} is A/r since

p(we+l,u) = M for all u E U and p(ve+l,vi) = M or M +,ri for all Wi E’ V. Furthermore,

note that p(u,wi) > R + 2111 - R = 2M 2 M + ri, for u E U and Vi E V, since,the (d + l)st

entry of u is R + 2M and the (d + l)st entry of vi is R or R - ri. Thus, for every ‘u. E U, the

radius associated with u in U U VU {we+l} is min{p(u, ve+l), p(u,&)lc E U} = A4. For all w E V,

the radius associated with ui in U U V U {TJ~+I} remains ri. Thus, U U V U {w+l} realizes G

in lRd+‘. I

Theorem 7 can be extended as follows.

THEOREM 8. Let G be a connected graph with vertex set V. Let S be a subset of V alld let H

be the subgraph of G induced by the set of vertices V \ S. Let @I, HZ,. . . , Ht be the connected

components of H of order 2 or more and let p be the number of isolated vertices of H or one,

whichever is greater. Thea

SIG(G) I ISI + max{rlogzpl,SIG(H1),...,SIG(Ht)}.

PROOF. Let V = {1,2,. . . , n} be the vertices of G and without loss of generality take S =

(1,. . ., s}. Since G is connected, we may further assume that the yertices of S have bken ordered

insuchawaythatvertexiisadjacenttosomevertexin(V\S)U{l,...,i-l}fori=1,2,...,~.

For i = 1,2,..., s, let Gi be the graph which consists of the components of order 2 or more of

the induced subgraph on the vertices V \ {i + 1,. . . , s}. By Theorem 7, SIG(G1) 5 1 + d where

d=max{[logzpl,SIG(H1),...,SIG(Ht)}.

Suppose that SIG(Gi) < i + d. Consider Gi+l. By Theorem 7, SIG(G.i+l) 5 1 + max{ [log, pil,

SIG(Gi)} where pi is the number of isolated vertices in the graph obtained when. vertex i + 1

is removed from Gi+l or one, whichever is greater. The assumption on the ordering of the

vertices of S, implies that pi 5 p. Thus, [log2pil < [log,pl <_ d. Since SIG(Gi) I: i + d,

SIG(Gi+l) 5 (i + 1) + d. The result now follows. I

The next corollary is the “independent set” analog of Corollary 4.

COROLLARY 9. Let G be a graph of order n witl? no isolated vertices. If G has an independent

set of size t > 1, then

SIG(G)<n-l-t+ [log,t].

PROOF. Suppose G has vertex set V = {1,2,. . .,n}andtheverticesinthesetW={1,2,...,t}

are independent. Without loss of generality, we may assume that G has no independent set

Sphere-of-Influence Graphs 1077

of size t + 1. Let v be a vertex of G not in W, and let S = V \ (W U {v}). The induced

subgraph with vertex set W U {v} consists of a star on t + 1 - q vertices and q isolated vertices

for some q E (0, 1,2,. . . ,t - 1)). Let p = q or one, whichever is greater. Note p 5 t, and hence,

~log,p~~~log,t].Ifq=t-1,thenSIG(K~,~)=1i~log~t~.Ifq<t-1,thenbyCorollary2,

SIG(KI,+~) 5 [log, t - ql 5 [log, tl. The result now follows from Corollary 8. I

We end this section by using Ramsey theory to give a bound on the SIG-dimension of a graph G

of order n with no isolated vertices. Corollaries 4 and 9 imply that if G has an independent set

of size t or has a clique of size t, then

SIG(G) < n - t + [log, tl .

Using Stirling’s approximation for factorials and an upper bound for Ramsey numbers (see for

example [lo]), it can be shown that if n > 4t, then G has an independent set of size t or a clique

of size t. These observations imply the following.

COROLLARY 10. Let G be a graph of order n 2 4 with no isolated vertices. Then

SIG(G) I n - t + [log, tl ,

where t = [log, n/21 a

4. COMPLETE MULTIPARTITE GRAPHS

In this section, we derive lower bounds on the SIG-dimension of complete multipartite graphs,

and in some cases determine the SIG-dimension. We begin by discussing two standard results in

graph theory.

As we have already discussed, in [9], it is shown that the biclique cover number of K, equals

[log, nl. Indeed, in [9], it is shown that in any covering of K, by bipartite graphs, the number

of bipartite graphs is at least [log, nl . A graph G is chordal (see [ll]) p rovided it contains no chordless cycles of length 4 or more. It

is well known that each interval graph is a chordal graph, and that the complement of a chordal

graph contains no chordless cycles of length 5 or more.

The following theorem is stated in the unpublished manuscript [5], but the proof given there

is incorrect. Our proof is motivated by the “correct” ideas in [5].

THEOREM 11. Let nl, 712,. . . ,nP 2 2 be integers and let no be a nonnegative integer. Set

ml = m2 = . . . = mno = 1. Then

PROOF. We first show that SIG(K,) 2 [log, nl. Let V = (~1,712,. . . , v,} be a set of vectors

in IWd that realizes K,. For C = 1,2 , . . . , d, define He to be the subgraph of Kn consisting of the

edges ij such that

p(wi,Vj) = vje’ 3 . - p)

We claim that He is bipartite. Suppose to the contrary that He contains a cycle of odd length.

Then there exist i, j, k such that

P(Wi, Wj) = ZJuzjej - ?$e, > 0 and p(v Q_) = #) - w(e) 3, 3 k >o.

It follows that p(w. Wk) 2 J[) - .@) = *I t k dwi, “j) + p(Wj, wk) 1 Ti + ?-k.

1078 E. BOYER et al.

This leads to the contradiction that vi and vk are not adjacent in SIG(V). Hence, Hr, . . . , Hd

are bipartite graphs which cover I(,, and thus, by [9], d 2 [log, n] . Now let G = G,,nz ,..., n,,,ml,mz ,..., m,,, , and assume that G is not complete. Thus, either p > 2,

or p = 1 and na 2 1. Suppose that V = {vi,j : 0 5 i 5 p, 1 5 j 5 ni} is a set of points in R”

that realizes G. We denote the kth entry of vi,j by Vet', and identify the vertices of G with the

vectors Vi,j.

Fore= 1,2,... , d, let Ze denote the interval graph determined by the intervals

where ri,j denotes the radius of the sphere associated with vi,j. Each Zt is a chordal graph, and

using the natural identification of intervals and vertices, G is the intersection of the graphs Ze

for != 1,2 ,..., d.

Note that G is the disjoint union of cliques I(n,, . . . , K,,,, and no isolated vertices. Let He

denote the complement of &. First note that if He has two edges from different connected

components of c’, then Ze contains a chordless cycle of length 4. Thus, since & is a chordal

graph, the edges of He lie in a single connected component of c (e = 1,2, . . . , d).

We claim that He is bipartite. Suppose to the contrary that He contains a cycle of odd length.

Since Ze is chordal, its complement contains no chordless cycle of odd length 5 or more. Thus, He

contains a cycle of length 3. Moreover, there exist i and distinct jr, jz, js such that, the intervals

are disjoint. Without loss of generality we may assume that

There exists a vertex Vi*,jf with p(Vi,j,, vi’,j’) = ri,jZ. Thus,

ril,jl 2 Tirj2.

Since G is a complete multipartite graph, i’ # i. Thus, the interval

(6)

(7)

( v!e). Z’J’

- r!e). v(e), + T., %‘J” %‘,J’ 2 93’ >

intersects each of the intervals in (6). In particular, ri’,jj > ri,jz, contrary to (7).

Hence, each such He is a bipartite graph. It follows that the graphs HI, Hz,. . . , Hd can be

partitioned into s + 1 parts where the it” part (for i = 1,2,. . . , s) consists of the collection of

graphs which cover Kni. Therefore, by [9], we conclude that d 2 C&, [log, nil. I

Combining Corollary 2 and Theorem 11, we can determine the SIG-dimension of each complete

multipartite graph which has at most one partite set of order 1.

COROLLARY 12. If n1,n2,. . . ,ns 12, then

SIG (Knl,n2 ,..., n,,~) = SIG (&,,n2 ,..., ,,) = f: b% nil .

More generally, we can determine the SIG-dimension

the number of partite sets of size one is relatively small.

i=l

of a complete multipartite graph when

Sphere-of-Influence Graphs 1079

COROLLARY 13. Let ml = mp = . . - = mn,, = 1 and nl, 722,. . . , n, 2 2 be integers. If no 5 C 2(b&‘G1-1) h w ere the sum runs over all ni > 2, then

SK (K, ,..., n,,ml ,..., m,,,) = 2 bg2 nil . i=l

PROOF. If no 5 1, then the result follows from Corollary 12.

Suppose that 1 < no 5 C 2 (P% nil-1) where the sum runs over all ni > 2. By Theorem 11,

SIG (f&1,...

We give a construction to show that

SIG (&, ,..., n,,m,, . . . . m,,,) i g r i=l

log, nil .

10th nil .

For i = 1,2,. . . , S, let iVi be a (1, -1)-matrix of size ni by [log2 nil with distinct rows. Let ti = 2(k3zn+l--l) for all i = 1,2 ,‘.‘, s. For each ni 2 3, let ?i be a (-O.l,O.l)-matrix of size ti

by ([log, nil - 1) with distinct rows and let Ti be the matrix which is obtained from !?i by adding

acolumnofzeros. Foralli,j= 1,2 ,..., S, i # j, let Fi,j be a (0,0.2)-matrix of size ti by flog, njl

where each entry in the last column is 0.2 and all other entries are zeros. In the case that ?zj = 2,

Fi,j is just a column of zeros. Let

N 0

L= 0

TI F2,t

_F 591

0

N2

0

FI,Z

T2

F 0

. . .

. . .

. . .

. . .

. . .

*.

. . .

O-

0

Ns

FL, ’ F2,s

T, _

Let i be the matrix whose rows are the first no + n1 + . . . + ns rows of L. Note that the

distance between two distinct rows of 2 is either 1, 2, or .2 depending on whether the vertices

are in different partite sets one of which has more than two vertices, in the same partite set, or

in different partite sets each with exactly one vertex. Hence, the radius of each row is either 1

or .2, depending on whether the row corresponds to a vertex in a partite set with more than one

vertex or not. It is now easily verified that the rows of i realize K,, ,,..., n,,,ml ,..., m,,, in

l@=I rk% nil. I

5. CONJECTURE AND EVIDENCE

In this section, we discuss the conjecture made in Section 1. In particular, we show that this

bound is best possible and give specific families of graphs which satisfy the conjectured bound.

We first show the conjecture is true for bipartite graphs.

COROLLARY 14. Let G be a bipartite graph of order n witJ1 no isolated vertices. Then SIG(G) <

Pn/31. PROOF. Suppose that G has bipartition X and Y with 1x1 = k and IYI = n - k, k 5 Ln/2J. By Corollary 9, SIG(G) 5 k - 1 + [log2(n - k)]. For 1 < k 5 [n/2J, the function f(k) = k - 1 + [log, (n - k)] achieves its maximum at k = [n/2]. Thus, SIG(G) 5 [n/2] - 1 + [log, [n/21 1.

It can be verified that [n/2] - 1 + [log2[n/211 5 [2n/31 for all n. I

We next show that the conjecture is true for graphs of order n < 8.

1080 E. BOYER et al.

COROLLARY 15. Let G be a graph of order n I 8 with no isolated vertices. Then

PROOF. If n = 2, then G consists of a single edge, and thus, SIG(G) = 1. If n = 3, then

Theorem 7 implies that SIG(G) 5 2. If n = 4, it is a simple exercise to show that G contains

either an induced path of length 2, or G is K 4. If G has an induced path of length 2, then by

Theorem 7, SIG(G) 5 2. If G is K4, then SIG(G) = 2. The result follows now for n = 5,6,7,8

from Theorem 7. a

The conjecture is also true for graphs of small maximum degree.

COROLLARY 16. Let G be a graph of order 11 with no isolated vertices and maximum degree at

most 3. Then

SIG(G) 2 $ . i 1

PROOF. The proof is by induction on n. By Corollary 15, we may assume that n 2 9 and proceed

by induction. Thus, we may assume that G is connected.

First assume that G has a vertex v of degree 1. Let w be the vertex of G which is adjacent

to V. Let S = {w}. Since the maximum degree of G is at most 3, G\ S has at most three isolated

vertices. Using the assumption that n 2 9, the inductive hypothesis and Theorem 8, we conclude

that

Next assume that G has a vertex v of degree 2. Let w1 and w2 be the vertices of G which

are adjacent to V, and let S = (WI, wz}. Since v is an isolated vertex of the graph G \ S, each

connected component of G \ S has at most n - 3 vertices. Since the maximum degree of G is at

most three, the graph G \ S has at most five isolated vertices. Using the assumption that n > 9,

the inductive hypothesis and Theorem 8, we conclude that ”

Finally, assume that G is regular of degree 3. Let IC be a vertex of G with neighbors u, v, and w

and let the neighbors of 21 be a and b. (Note: {a, b} and {v, w} may intersect nontrivially.) Let

S = {a, b, v, w}. Then G \ S is not connected. Let p be the number of isolated vertices of G \ S

or one, whichever is greater, and let HI, Hz,. . . , Ht be the nontrivial components of G \ S. Note

that one of the nontrivial components, say HI, consists of the edge joining x and u.

Since each isolated vertex of G \ S is adjacent to three vertices in {a, b, v, w}, G is regular of

degree 3, and G has at least nine vertices, we conclude that p 5 3. In particular,

1st + [log,pl I 4 + 2 I $ . 11

Since n 2 9 and p 5 3, there is at least one other nontrivial component. Hence, each nontrivial

component contains at most n - ISI - 2 vertices. Thus, by induction, i

ISI + SIG(&) 5 ISI + p (n - ISI - 2)1 I 1: + i (ISI - 4)] < [$I , Theorem 8 now implies that SIG(G) < [2n/31, and the corollary follows by induction. I

Let K(3,s) be the complete multipartite graph with 3s vertices where each partite set is of

size three. Then by Corollary 12, the SIG-dimension of K(3, s) is 2s, and hence, the conjectured

bound is best possible. Furthermore, graphs which are “close” to this ‘complete multipartite

graph also satisfy the conjectured bound. We present these ideas in the next threk propositions.

Throughout the remainder of this section, we assume that the vertices of K(3, s) are labeled

Wl,l,W1,2,W1,3,.-* ,Ws,lr’Ws,Z,Ws,3~

where wi,l, wi,2, wi,3 are the partite sets for i = 1,2,. . . , s.

Sphere-of-Influence Graphs 1081

PROPOSITION 17. Let G be a graph of order 3s, s 2 2, which is obtained from K(3, s) by adding

some edges. Then SIG(G) 2 [2(3~)/3].

PROOF. If G is obtained by adding edges to K(3, s), then ?? consists of disjoint subgraphs, each with at most three vertices. Since a graph with three vertices has a distinguishing biclique cover

of size two, the result follows from Theorem 1. 1

Pno~osr~ro~ 18. Let G be a graph with 3s vertices, s 2 2, which is obtained from a K(3, s) by

removing the edges of a matching. Then SIG(G) 5 [2(3~)/3].

PROOF. Since TQ~, w~,~, wi,3 are mutually adjacent in ??, each set of independent vertices of ?? has at most e vertices. It follows that E has 2s vertices such that every edge of ?? is incident to at least one of these vertices. The stars at these vertices give a distinguishing biclique cover of ??

consisting of 2s bicliques. I

I&OPOS~T~ON 19. Let G be a graph of order.3s + 1, s 2 2, which is obtained from K(3,s) by add& one new vertex v which is adjacent to at least one vertex of K(3, s). Then SIG(G) <

r2(34/31. PROOF. Let, M be the 3s by 2s matrix which is the direct sum of s copies of the following 3 by

2 matrix -1 -1

1 1 1 -1 . 1 1

Note that K(3, s) is realized by the rows of M. Let v = (or,r, wr,z, . . . , VJ~,I, 21,,2) where

1

(O,O), if ‘U is adjacent to all 3 of wi,r,wi,s, wi,s,

(V&l, q2) = (0, -l), if v is adjacent to 2 of ‘wi,r, WQ, wi,s,

(-l,O), if 21 is adjacent to 1 of wi,r,wi,z,wi,s,

(-1, l), if u is adjacent to none of wi,l,wi,2, wi,3.

Since v is adjacent to at least one vertex of K(3, S) and s 2 2, the rows of M together with the vector v realize a graph which is isomorphic to G. I

We conclude this section by using Theorem 8 to obtain a bound for all realizable graphs.

THEOREM 20. Let G be a graph of order n with no isolated vertices. Suppose G has maximum degree A. Then

SIG(G) I & . 1 1 PROOF. The proof will be by induction on n. If A = 1, then G is a collection of &joint edges,

and hence, SIG(G) = 1, and the inequality holds. By Corollary 15, the result holds for graph of order 8 or less. Thus, we may assume n 2 9 and that the inequality holds for all graphs with less than n vertices.

Let x be a vertex of G of degree A and let S = {xl, x2,. . . , xa} be the neighbors of z in G. Then the graph G \ 5’ is a disconnected graph. Let HI, Hz,. . . , Ht be the connected components of G \ S of order 2 or more and let p be the number of isolated vertices of G \ S or one, whichever is greater. Then

SIG(G) 5 A + max{ [log,pl, SIG(Hr), . . . , SIG(H,)}.

Since G \ S is disconnected and 1st = A, we have

A + SIG(Hi) 5 A + &$n-(A+l))

fori = 1,2,... , t. Now consider A+ [log, pl . Since IS/ = A, p L n-A. Therefore, A + [log, pl 5 A + rlog2(n - A)l. Let f(x) = log,(n -2) - (x/x + l)n+x. It is easily verified that f(n - 1) = 0, and f(2) 5 0 and f(x) is concave up. Thus, f(x) I 0 for 2 I x < n, and we conclude that A + [log2(n - A)1 I [An/A +,ll. The result now follows. I

1082 E. BOYER et al.

1.

2. 3.

4.

5. 6.

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