The interval number $i( G )$ of a simple graph $G$ is the smallest number $t$ such that to each vertex in $G$ there can be assigned a collection of at most $t$ finite closed intervals on the real line so that there is an edge between vertices $v$ and $w$ in $G$ if and only if some interval for $v$ intersects some interval for $w$. The well known interval graphs are precisely those graphs $G$ with $i ( G )\leqq  1$. We prove here that for any graph $G$ with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil $. This bound is attained by every regular graph of degree $d$ with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil $ for graphs on $n$ vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor $ for graphs with $e$ edges

B. West

Douglas

Jerrold R. Griggs

English

The interval number i(G) of a simple graph G is the smallest number such that to each vertex in G there can be assigned a collection of at most finite closed intervals on the real line so that there is an edge between vertices v and w in G if and only if some interval for v intersects some interval for w. The well known interval graphs are precisely those graphs G with i(G)=&lt;I. We prove here that for any graph G with maximum degree d, i(G) &lt;- [1/2(d + 1)]. This bound is attained by every regular graph of degree d with no triangles, so is best possible. The degree bound is applied to show that i(G) &lt;- [1/2n] for graphs on n vertices and i(G)&lt;- [/J for graphs with e edges

Douglas B. West

CiteSeerX

Extremal values of the interval number of a graph

Griggs, Jerrold R.

West, Douglas B.

Caltech Authors - Main

SIAM J. ALG. DISC. METH.
Vol. 1, No. 1, March 1980
1980 Society for Industrial and Applied Mathematics
0196-5212/80/0101-0001 $01.00/0
EXTREMAL VALUES OF THE INTERVAL NUMBER
OF A GRAPH*
JERROLD R. GRIGGS" AND DOUGLAS B. WEST
Abstract. The interval number i(G) of a simple graph G is the smallest number such that to each vertex
in G there can be assigned a collection of at most finite closed intervals on the real line so that there is an edge
between vertices v and w in G if and only if some interval for v intersects some interval for w. The well known
interval graphs are precisely those graphs G with i(G)=<I. We prove here that for any graph G with
maximum degree d, i(G) <- [1/2(d + 1)]. This bound is attained by every regular graph of degree d with no
triangles, so is best possible. The degree bound is applied to show that i(G) <- [1/2n] for graphs on n vertices
and i(G)<- [/J for graphs with e edges.
1. Introduction to interval numbers. We begin by discussing earlier work on
interval graphs and boxicity in order to motivate the definition of interval numbers. A
simple bound on the interval number given the numbers of edges and of vertices of a
graph is presented along with results on the interval number of some basic graphs. In the
next section we prove our main result, which gives the best-possible upper bound on the
interval number of a graph given its maximum degree. This is applied to obtain an upper
bound on the interval number given only the number of vertices. We conclude by listing
several interesting open problems.
Interval graphs are simple undirected graphs G with the property that there exists a
collection of finite closed intervals on the real line such that an interval [ao, bo] is
assigned to each vertex v in G and such that the intervals assigned to two vertices v and
w in G intersect each other if and only if they are joined by an edge in G. Interval graphs
have been studied extensively and can be nicely characterized [1], [3], [4], [5]. They
have important applications to various problems of scheduling, allocation, and
sequencing.
It is natural to try to extend this idea of representing graphs by intersections of
intervals to all graphs. For even some simple graphs, such as the n-cycles Cn (n > 3), are
not interval graphs. One approach, taken by Roberts [7], [8], is to go to higher
dimensional intervals: define the boxicity of a graph G to be the smallest integer such
that G can be represented by the intersections of t-dimensional "boxes" which have
their edges parallel to the coordinate axes. That is, to each vertex v is assigned an
ordered collection of finite closed intervals
([ao,1, by, l], [a,2, bo,2], [a,,, b,t]),
and two vertices v and w are joined by an edge in G if and only if [ao, i, b,i] intersects
[aw,i, bw,i] for all i. In these terms, interval graphs are precisely the graphs with boxicity
at most 1.
Here we present a different approach to extending interval representations to all
graphs. We expect that this approach will be useful in dealing with certain scheduling
and allocation problems, such as traffic light assignments [9] and radio frequency
assignments [2]. Rather than going to higher-dimensional intervals, we allow each
vertex to be represented by a collection of several intervals. Define the interval number
* Received by the editors January 22, 1979.
"
Department of Mathematics, California Institute of Technology, Pasadena, California 91125.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139.
2 JERROLD R. GRIGGS AND DOUGLAS B. WEST
of a graph G, denoted i(G), to be the smallest integer t->0 such that there can be
assigned to each vertex in G a collection of at most t finite closed intervals so that there
is an edge in G between vertices v and w if and only if some interval for v intersects
some interval for w. This definition is due to R. McGuigan [6].
i(G) exists for any graph G: An interval representation of G is obtained by taking a
pair of overlapping intervals, one labelled v and the other w, for each edge {v, w} in G.
These pairs of intervals are to be separated from each other. Of course, this construction
will not achieve the value i(G) in general.
Interval graphs are precisely those graphs with i(G)-_< 1. Only for graphs with no
edges does i(G)= 0. Complete graphs are interval graphs, so i(Kn)= 1. To represent
Kn, just stack up n intervals, one per vertex, so that their mutual intersection is
nonempty. The cycles Cn, n > 3, are not interval graphs, (Cn) 2 as the representation
in Fig. 1 shows for C4.
2
2 4
3
FIG.
Using the Lekkerkerker-Boland forbidden subgraph characterization of interval
graphs [5], it is straightforward to show that for trees T, (T) 1 if and only if T contains
no induced subgraphs of the form shown in Fig. 2. Otherwise, i(T)= 2. See [10] for
details.
FIG. 2
Next we consider arbitrary graphs which contain no triangles. We present a simple
proof of a useful lower bound on i(G) given the number of edges and the number of
vertices in G. Here [x denotes the least integer no smaller than x, and [x denotes the
integer part of x.
THEOREM 1. Let G be a simple graph on n vertices and e > 0 edges which contains no
K3. Then i(G)>-_ [(e + 1)/n].
Proof. As e > 0, we have (G) -> 1. Suppose we are given an interval representation
I for G which attains the bound i(G), i.e., uses no more than i(G) intervals per vertex.
As G contains no g3, no three intervals in I may share a point. For each edge {v, w} in
G, there must be a stretch of points on the real line where a v-interval overlaps a
w-interval. At the right end of such a stretch, one of the two intervals must end. It
follows that there must be at least e + 1 intervals in /. Thus some vertex must be
represented by at least [(e + 1)/n] intervals, so that i(G) >- [(e + 1)/n]. !-!
INTERVAL NUMBERS 3
We thus derive a lower bound on the interval numbers of complete bipartite
graphs:
COROLLARY.
+ 1]i(gm,n)! rn / n I"
We have constructed interval representations which show that this lower bound is
actually the correct value of i(Km,n) in various special cases. But Trotter and Harary
[10], working independently of us, have proposed the same definition of i(G) and have
come up with a construction for all rn and n of interval representations of K,,.n using at
most [(mn + 1)/(rn + n)] intervals per vertex to show that this lower bound is always the
actual value of (K,...).
We have some results on i(G) for complete p-partite graphs with p > 2 which we
are currently trying to improve and plan to discuss in another paper.
2. The degree bound. Now we come to the main result of this paper, the
best-possible upper bound on i(G) for graphs G which have degree at most d at each
vertex. Note that an upper bound is what is interesting; the lower bound is just 1 for all
d > 0, because (Kd+ 1) 1. The construction following the definition of (G) established
an upper bound of d on (G). Here we shall lower this bound to [1/2(d + 1)] and show that
it is actually attained by some graphs. The interval representation to attain this upper
bound is simple to construct for a given graph, and it has some other nice properties. We
apply this result in the next two sections to obtain upper bounds on i(G) when G has
a given number of vertices or edges. For convenience let d(v) denote the degree of
vertex v.
THEOREM 2. If G is a graph with d maxv d(v) > 0, then i(G)<= [1/2(d + 1)].
Proof. For any graph G with d as above we must give an interval representation for
G using at most d intervals per vertex in order to prove the theorem. We do this by
induction on the number n of vertices of G using this stronger induction hypothesis:
(,) For any graph G on n vertices and any vertex v in G there is an interval
representation of G in which the leftmost interval is a v-interval and in
which, for each vertex w in G, there are at most [1/2(d(w) + 1)] w-intervals.
An interval [a, b is leftmost (respectively, rightmost) if for any other interval [c, d] in
the representation, a < c(b > d).
Hypothesis (.) holds trivially for n 1. So assume that G has n > 1 vertices and
that (.) holds for all graphs on fewer than n vertices. Let v be any vertex in G. We now
construct an interval representation satisfying (.).
Suppose first that there is a circuit C1 passing through v in G. By circuit we mean a
path which begins and ends at v without repeating edges and without passing through
any vertex twice. Say C1 v, Wl, w2, , Wk, V lists the vertices in C1 in order, where
k -> 2. Figure 3 shows an interval representation of the edges in Ca. Now remove these
edges from G (but not the vertices). Suppose there remains another circuit C2 through
v. Then represent each edge in C2 using the same idea as for Cx, except the v-interval on
the right for C1 is used as the v-interval on the left for representing C2. Continue this
wl u
U W2 Wk
FIG. 3
4 JERROLD R. GRIGGS AND DOUGLAS B. WEST
procedure of representing and deleting the edges in circuits through v until no more
circuits pass through v.
In the case that there is no such circuit C1 passing through v, just put down a single
v-interval. So, in general, if we remove rn circuits through v, the number of v-intervals
used is precisely rn + 1, and the left-most and rightmost intervals are v-intervals. In
removing these circuits, the degree of each vertex w v is reduced by twice the number
of w-intervals used in the representation.
If v now belongs to no edges, apply (.) by induction to the rest of G to obtain a
representation which satisfies the degree bound in (.) at each vertex. Otherwise,
suppose that there are p > 0 vertices adjacent to v in G, which we call u 1, u2, , up.
Since no circuits pass through v now, the vertices ui lie in distinct components if v is
deleted from G. Let Gi be the component containing ui. By induction there is an interval
representation Ii of G in which ui is leftmost and in which the number of intervals for
each vertex is bounded according to (.).
Put 11 to the right of the intervals used to represent the circuits of G so that the
leftmost u 1-interval in 11 overlaps the rightmost v-interval. This represents the edge
{v, Ul} and all edges in G1. For p > 1, add [p/2] additional v-intervals to the right of the
intervals used thus far. Reverse the order of representations 12, 14, I6, so that there
are intervals for u2, u4, u6, which are rightmost in their representations. Then insert
the I in the representation of G so that I2 is to the left of the leftmost new v-interval, I3
is to its right, I4 is to left of the second new v-interval, and so on. The extreme ui-interval
in li should overlap the v-interval. Figure 4 shows the construction. To complete the
construction, represent any remaining edges, by induction on (.), with intervals to the
right of all the other intervals.
C C2 Cm II 12 I3
ul u2 u3,--"--
v v v v v
FIG. 4
We have now represented all the edges of G, and no others. A v-interval is
leftmost. By counting the intervals used, it follows that not more than [1/2(d(w)+ 1)]
intervals are used for any vertex w in G. Thus (.) is satisfied, and the theorem is
proven. I3
That this bound is best possible follows from this result:
COROLLARY. For any regular graph G of degree d containing no K3,
i(G) [1/2(d + 1).
Proof. Suppose G is a regular graph of degree d containing no K3, and let n be the
number of vertices of G. G has exactly nd edges, so by Theorem 1,
i(G)>= [(1/2nd+l)/n] [1/2(d + 1)],
and this is just the upper bound on i(G) in Theorem 2. 71
Two important examples of such graphs are Kd,d and Qd, the d-dimensional cube.
A strong property of the representation in the proof of Theorem 2 that may be
useful in some applications is that it has depth two: no three intervals overlap on the real
line.
3. The vertex bound. Another extremal problem of interest is this: Among all
graphs G on n vertices, how large can i(G) be? Since d =< n 1, it follows from Theorem
INTERVAL NUMBERS 5
2 that i(G)<= [1/2n]. As an application of Theorem 2 we present here a nice interval
construction to improve this bound on i(G) to [1/2n].
THEOREM 3. If G has n vertices, then i(G)<= [1/2n ].
Proof. The proof is by induction on n. It is certainly true for n <= 3. Now suppose G
has n > 3 vertices. We prove i(G) =< [In in two cases, depending on whether or not G
has a triangle.
First suppose G contains some triangle, on vertices T- {u, v, w}. The intervals
shown in Fig. 5 represent the edges in T and have the additional property that for any
subset of T there is a stretch of the line in which intervals for precisely this subset
W
FIG. 5
overlap. Hence we can represent all edges between T and G- T (the vertices outside
T) using at most one interval per vertex in G- T: For example, if a vertex x in G- T
neighbors u and w, put a small x-interval inside the interval where the u- and
w-intervals overlap, but no v-interval does. At most two intervals are used for each
vertex in T in taking care of all edges involving T. Now, by induction, represent all edges
between vertices in G-T, using at most [1/2n]- 1 intervals per vertex. Place these
intervals away from those involving T to complete the construction.
It remains to consider graphs G with no triangle. By Theorem 2, i(G) <= [1/2n] holds
provided that [1/2(d + 1)] -<_ [1/2n], or, equivalently, d <-2 [1/2n] 1. Thus in the remaining
case it suffices to assume that G contains some vertex v of degree at least n. Let W be
the set of vertices adjacent to v. There are no edges in W because G has no triangles.
Let X be the set of vertices outside W t.J {v}. To represent all edges incident on W, take
a long interval for each of the vertices in X LI {v}, no two intersecting, and for each edge
{w, y}, with w W and y X LI {v}, put a small w-interval inside the y-interval. (See Fig.
6.) At most n intervals are used for each w W because [X{v}[<-1/2n. The only
remaining edges involve pairs of vertices in X and can be represented, by induction,
using at most [1/2[X[] intervals per vertex in X. This represents G with at most [31-n]
intervals per vertex.
Trotter and Harary [10] independently discovered the same [1/2n] bound on i(G).
The constructiori given here is simpler. How good is this bound? The balanced complete
bipartite graphs, Kt,,/21.,,/:z, show that i(G) can get at least as large as [1/4(n + 1)]. This
agrees with [1/2n for n < 7. At n 7 it is not difficult to prove that (G) can be at most 2,
so the [1/2n]-bound is not always best possible. It is natural to conjecture that these
graphs K t,,/z.;,,/z are extremal among all graphs G on n vertices, just as they were for
graphs of maximum degree d. That is, the best possible upper bound on i(G) should be
[1/4(n + 1)]. One of us (Griggs) has recently succeeded in showing this, but owing to the
length and complexity of the proof, it will appear elsewhere [13].
4. The edge bound. How large can the interval number of a graph with e edges
get? Theorem 2 can again be applied to give an upper bound.
THEOREM 4. If G has e edges, then i(G)<- [x/].
Proof. The theorem holds trivially if e =< 1, so assume that G has e > 1 edges and
that the theorem holds for all graphs with fewer than e edges. Let k [x/]. If d < 2k,
then i(G)<-k by Theorem 2. So assume that d>=2k and let v be a vertex of degree d.
Represent all edges containing v by a long v-interval overlapped by a small w-interval
6 JERROLD R. GRIGGS AND DOUGLAS B. WEST
for each neighbor w of v (see the v-interval in Fig. 6). This requires at most one interval
per vertex in G. The remaining e d < (k 1)2 edges in G can be represented using at
most k- 1 intervals per vertex by induction. I-1
W-intervals W-intervals W-intervals
V Xl X2
FIG. 6
intervals for edges
inX
This argument can be refined to obtain the slightly stronger result that i(G) <= [x/J.
It can be shown that i(G)-< 2 for e 9, so this upper bound [x/] is not best possible.
The graphs K2,,.2,,, m 1, 2, 3,. ., show that i(G) can get at least as large as 1 + [1/2x/J.
We conjecture that this is also an upper bound, which would be best possible.
5. Areas requiring turther study. Applications will motivate the study of other
problems related to interval numbers. We propose the following:
1. Give a forbidden subgraph characterization of the graphs with interval number
at most k, where k _-> 2.
2. Interval numbers minimize the maximum number of intervals used for any
vertex in representing G. One could instead seek to minimize the total number
of intervals required in a representation.
3. Representations could be restricted to being of depth at most r by not allowing
any r + 1 intervals to share a point. What can be said about the "depth r interval
number"?
4. Rather than intervals, one can use circular arcs to represent vertices and ask for
a circular interval number ic (G). This means that we allow a single interval to go
to+ and come back from -, so that (-o, a Ib, ), with a < b, counts as a
single circular interval, ic(Cn) 1 < i(Cn) 2 for n > 3. Graphs with it(G) <= 1
are known as circular-arc graphs [11], [12]. What is the behavior of i(G)?
It should be similar to i(G) since for all graphs, i(G) >- ic(G) >- i(G)- 1.
Acknowledgments. We are indebted to Fred Roberts for introducing us to interval
graphs and for bringing the work of Trotter and Harary to our attention; to Robert
McGuigan for proposing the study of interval numbers; and to Daniel J. Kleitman for
making some valuable suggestions. This work originated at the NSF-CBMS Regional
Conference in Graph Theory at Colby College, June, 1977.
REFERENCES
[1] D. R. FULKERSON AND O. A. GROSS, Incidence matrices and interval graphs, Pacific J. Math., 15
(1965), pp. 835-855.
[2] E. N. GILBERT, Mobile radio lrequency assignments, unpublished technical memorandum, Bell
Telephone Laboratories, 1972.
[3] P. C. GILMORE AND A. J. HOFFMAN, A characterization ofcomparability graphs and ofinterval graphs,
Canad. J. Math., 16 (1964), pp. 539-548.
[4] G. HAJOS, Uber eine Art yon Graphen, Internat. Math. Nachr., 47 (1957), p. 65.
[5] C. B. LEKKERKERKER AND J. CH. BOLAND, Representation ofa finite graph by a set ofintervals on the
real line, Fund. Math., 51 (1962), pp. 45-64.
[6] R. McGUIGAN, presentation at NSF-CBMS Conference at Colby College, June, 1977.
[7] F. S. ROBERTS, On the boxicity and cubicity ofa graph, Recent Progress in Combinatorics, W. T. Tutte,
ed., Academic Press, New York, 1969, pp. 301-310.
INTERVAL NUMBERS 7
[8] F. S. ROBERTS, Graph Theory and its Applications to Problems of Society, lectures presented at
NSF-CBMS Conference at Colby College, June 1977, Society for Industrial and Applied Mathe-
matics, Philadelphia, 1978.
[9] K. E. STOFFERS, Scheduling of traffic lights--A new approach, Transportation Research, 2 (1968), pp.
199-234.
[10] W. T. TROTTER, JR., AND F. HARARY, On double and multiple interval graphs, J. Graph Th., to
appear.
[11] A. C. TUCKER, Characterizing circular-arc graphs, Bull. Amer. Math. Soc., 75 (1970), pp. 1257-1260.
[12],Matrix characterizations of circular-arc graphs, Pacific J. Math., 39 (1971), pp. 535-545.
[13] J. R. GRIGGS, Extremal values of the interval number of a graph, II, Discrete Math., to appear.


Extremal Values of the Interval Number of a Graph

A characterization ofcomparability graphs and ofinterval graphs,

characterizations of circular-arc graphs,

Characterizing circular-arc graphs,

Extremal values of the interval number of a graph, II, Discrete Math.,

Graph Theory and its Applications to Problems of Society, lectures presented at NSF-CBMS Conference at Colby College,

HOFFMAN,A characterization ofcomparability graphs and ofinterval graphs,

Incidence matrices and interval graphs,

Matrix characterizations of circular-arc graphs,

Mobile radio lrequency assignments, unpublished technical memorandum, Bell Telephone Laboratories,

Extremal Values of the Interval Number of a Graph

Abstract

Similar works

Full text

Available Versions

CiteSeerX

Caltech Authors - Main