On the profile of the corona of two graphs
Yung-Ling Lai
a, Gerard J. Chang
b,∗aGraduate Institute of Computer Science and Information Engineering, National Chia-Yi University, Chiayi, Taiwan bDepartment of Mathematics, National Taiwan University, Taipei 106, Taiwan
Received 3 November 2003; received in revised form 10 December 2003 Communicated by S. Albers
Abstract
The concept of profile, together with bandwidth, originates from handling sparse matrices in solving linear systems of equations. Given a graph G, the profile minimization problem is to find a one-to-one mapping f : V (G)→ {1, 2, . . . , |V (G)|} such thatv∈V (G)maxx∈N[v](f (v)− f (x)) is as small as possible, where N[v] = {v} ∪ {x: x is adjacent to v}. This paper
studies the profile of the corona G∧ H of two graphs G and H . In particular, bounds for the profile of the corona of two graphs are established. Also, exact values of the profiles of coronas G∧ H are obtained when G has certain properties, including when G is a caterpillar, a complete graph or a cycle.
2003 Elsevier B.V. All rights reserved.
Keywords: Profile; Corona; Numbering; Interval graph; Caterpillar; Complete graph; Cycle
1. Introduction
The concept of profile, together with bandwidth, originates from handling sparse matrices in solving linear systems of equations, see [2,8,11,12,17] and their references. This problem can be re-formulated as the following problem in graphs.
A numbering (or labeling or layout) of a graph G is a one-to-one mapping f from V (G) onto{1, 2, . . .,
|V (G)|}. For a numbering f , the profile-width of a
vertex v is defined as
wf(v)= max x∈N[v]
f (v)− f (x),
* Corresponding author. Supported in part by the National
Science Council under grant NSC92-2115-M002-015. E-mail addresses: yllai@mail.ncyu.edu.tw (Y.-L. Lai), gjchang@math.ntu.edu.tw (G.J. Chang).
where N[v] = {x ∈ V (G): x = v or xv ∈ E(G)} is the
closed neighborhood of v. The profile of a numbering f of G is
Pf(G)=
v∈V (G)
wf(v)
and the profile (or skyline) of G is
P (G)= minPf(G): f is a numbering of G
.
A numbering f is called a profile numbering of G if
Pf(G)= P (G).
The profile minimization problem has many equiv-alent definitions, such as interval graph completion [1,20] and graph searching [5]. In this paper, we are more concerned about the interval graph completion. A graph G= (V, E) is an interval graph if we can as-sociate each vertex v with a (closed) interval Ivin the
real line such that two different vertices x and y are
0020-0190/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2003.12.004
adjacent if and only if Ix∩ Iy= ∅. The interval graph
completion problem is for a graph G finding a super-graph of G (which is a super-graph Gwith V (G)= V (G)
and E(G)⊆ E(G)) of minimum size that is an
inter-val graph.
Proposition 1 [1,20]. For any graph G the profile of G
is equal to the smallest number of edges in an interval supergraph of G.
The interval graph completion problem is known to be NP-complete even for edge graphs (see [7]). On the other hand, there are linear-time algorithms for cographs [5,14] and an O(n1.722)-time algorithm for
trees of n vertices [14,15]. Because of its important applications, quite a few approximation algorithms have also been developed, see [13,23,28–30]. We say that the profile minimization problem is solved for a class of graphs if either a polynomial-time numbering algorithm which achieves the profile for each graph in the class is provided or a formula of the exact value is given for each graph in the class. The profile minimization problem is solved only for a few classes of graphs (most are graphs obtained from two graphs applying some graph operations), see [9,16–18,20,21,24]. Lai and Williams [19] provide a survey of recent works. The current paper is to study the profile of the corona of two graphs, which was introduced in [6].
Definition. Given graphs G and H with n and m
ver-tices, respectively, the corona of G with respect to
H is the graph G∧ H with vertex set V (G ∧ H )
= V (G) ∪ {n distinct copies of V (H ) denoted V (H1),
V (H2), . . . , V (Hn)} and edge set E(G∧H ) = E(G)∪
{n distinct copies of E(H ) denoted E(H1), E(H2),
. . . , E(Hn)} ∪ {(ui, v): ui∈ V (G), v ∈ V (Hi)}.
Fig. 1 shows P3∧ P4.
Graph parameters such as bandwidth, edgesum, and domination number have been studied on coro-nas of graphs, see [3,4,10,27,31]. Also, the incidence
Fig. 1. P3∧ P4.
coloring of corona graphs has been studied by several researchers, see [22,25,26]. In this paper we establish bounds for the profile of the corona of two graphs. Ex-act values of the profiles of coronas G∧ H are deter-mined when G has certain properties, including when
G is a caterpillar, a complete graph or a cycle.
2. Bounds for general graphs
In this section we establish bounds for the profile of the corona of two general graphs. For convenience, we introduce the following terminology.
An interval-labeling of a graph G (which is not necessary an interval graph) is a mapping f on
V (G) that maps each vertex v to a (closed) interval
f (v) such that xy∈ E(G) implies f (x) ∩ f (y) = ∅.
Let If(G) denote the number of unordered pairs {f (x), f (y)} with x = y for which f (x) ∩ f (y) = ∅;
and I (G) denoted the minimum value of If(G),
where f is taken over all interval-labellings of G. An interval-labeling f is called I -optimal if If(G)=
I (G). According to Proposition 1, P (G)= I (G) for
any graph G. We will use P (G) and I (G) inter-changeably.
In the above definition, we may assume that f is
canonical (i.e., the 2|V (G)| endpoints of the intervals f (v) are all distinct) as this does not affect the value
of I (G). In particular, each interval f (v) is non-trivial (i.e., has a positive length) and the intersection of any two intersecting intervals f (x) and f (y) is also non-trivial. We assume that all interval-labellings are
canonical throughout this paper.
Suppose f is an interval-labeling of a graph G. For any vertex v, denoted by Mf(v) the minimum number
of intervals f (x) intersecting a non-trivial interval Jv
with Jv∩ f (v) = ∅, where x is taken over all vertices
in G and Jv is taken over all possible intervals with
Jv∩ f (v) = ∅. Notice that Mf(v) is a finite number
even if there are infinitely many possible Jv as there
are only finitely many f (x). Denote by Mf(G) the
sum v∈VMf(v), and M(G)= min{Mf(G): f is
an interval-labeling of G}. An interval-labeling f is called M-optimal if Mf(G)= M(G).
In the above definition, we may assume that Jvis a
non-trivial subinterval of f (v) and all Jvs are pairwise
disjoint, as this does not affect the value of M(G). We are now ready to establish a lower bound for the profile P (G∧ H ) in terms of P (G), P (H )
and M(G). We assume that V (G)= {u1, u2, . . . , un}
and the copy of H corresponding to ui is Hi with
vertex set V (Hi)= {vi1, vi2, . . . , vim} for 1 i n.
Lemma 2. If G and H are two graphs of orders n and
m, respectively, then P (G∧ H )
= minIg(G)+ nP (H ) + Mg(G)m:
g is an interval-labeling of G. (1)
Proof. Suppose f∗ is an I -optimal interval-labeling of G∧ H . Let gbe the restriction of f∗on V (G) and
hi the restriction of f∗on V (Hi) for 1 i n. Then
P (G∧ H ) = If∗(G∧ H ) Ig(G)+ n i=1 Ih i(Hi)+ m j=1 n i=1 bij,
where bij is the number of intervals g(uk)s that
intersect the interval hi(vij). Notice that the inequality
may possibly be strict if f∗(vij) intersects f∗(vij)
for some i= i. As hi(vij)∩ g(ui)= ∅, we may
choose hi(vij) as Jui in the definition of Mg to get
bij Mg(ui). Then P (G∧ H ) Ig(G)+ n i=1 P (Hi)+ m j=1 n i=1 Mg(ui)
right-hand side of Eq. (1).
On the other hand, suppose g∗ is an interval-labeling of G that attains the minimum in the right-hand side of Eq. (1). For each ui in V (G) there is
an interval Jui with Jui ∩ g∗(ui)= ∅ that intersects
Mg∗(ui) intervals g∗(uk)s. Without loss of generality,
we may assume that each Jui is a non-trivial
subin-terval of g∗(ui) and all Juis are pairwise disjoint. For
each i we may choose an I -optimal interval-labeling
h∗i of Hi such that each h∗i(vij) is a subinterval of Jui.
Define the mapping fon V (G∧ H ) by
f(x)=
g∗(x), if x∈ V (G);
h∗i(x), if x∈ V (Hi).
It is clear that fis an interval-labeling of G∧ H with
P (G∧ H ) If(G∧ H ) = Ig∗(G)+ n i=1 Ih∗i(Hi)+ m j=1 n i=1 bij,
where bij is the number of intervals g∗(uk)s that
intersects the interval h∗i(vij). Then bij Mg∗(ui) and
so P (G∧ H ) Ig∗(G)+ n i=1 P (Hi)+ m j=1 n i=1 Mg∗(ui) = Ig∗(G)+ nP (H ) + Mg∗(G)m.
Both inequalities prove the theorem. ✷
Consequently, we have the following lower bound for the profile of the corona of two graphs.
Theorem 3. If G and H are two graphs of orders n
and m, respectively, then
P (G∧ H ) P (G) + nP (H ) + M(G)m.
Proof. The theorem follows from Lemma 2 and the
fact that Ig(G) P (G) and Mg(G) M(G) for any
interval-labeling g of G. ✷
The minimization in Eq. (1) depends not only on
G but also on the value m. It is also the case that
the minimization is attained for the “mixed” value of both Ig and Mg, rather than individual Ig and Mg.
However, if there is a g∗ that attains the minimum individually for Igand Mgat the same time, then the
lower bound in Theorem 3 is in fact the exact value.
Theorem 4. If G has an interval-labeling g∗ that is both I -optimal and M-optimal, then
P (G∧ H ) = P (G) + nP (H ) + M(G)m.
Proof. The theorem follows from Lemma 2 and the
fact that g∗attains the minimum of the right-hand side of Eq. (1). ✷
Next, we establish an upper bound for the profile of the corona of two graphs in terms of their profiles.
Theorem 5. If G and H are two graphs of orders n
and m, respectively, then
Proof. Let g be a profile numbering of G and h be
a profile numbering of H . Define a numbering f of
G∧ H as follows: f (x)= (m+ 1)g(x), if x ∈ V (G); (m+ 1)(g(y) − 1) + h(x), if x∈ V (H ) and (x, y) ∈ E(G ∧ H ). Then P (G∧ H ) Pf(G∧ H ) (m + 1)P (G) + nP (H )+ nm. ✷
The upper bound in Theorem 5 is tight as illustrated by En ∧ H , where En is the complement of the
complete graph Kn. Note that En∧ H is n separate
copies of K1∧H whose profile P (K1∧H ) = P (H )+
m, and then P (En∧ H ) = n(P (H ) + m) = (m +
1)P (En)+ nP (H ) + nm.
3. Exact values
This section establishes exact values of the profiles of coronas G∧ H by means of Theorem 4. These include when G is a caterpillar, a complete graph or a cycle, for which the values of P (G) and M(G) are also determined.
We first consider the case when G is a caterpillar. A caterpillar is a tree from whom the removing of all leaves results a path (possibly empty). More precisely, suppose n= r +1<i<rsi, where r 2
and each si 0. A caterpillar with the parameters
(n; r; s2, s3, . . . , sr−1) is the tree T with
vertex set V (T )= {x1, x2, . . . , xr}
∪
1<i<r
{yi1, yi2, . . . , yisi}
and
edge set E(T)= {x1x2, x2x3, . . . , xr−1xr}
∪
1<i<r
{xiyi1, xiyi2, . . . , xiyisi}.
Theorem 6. If T is a caterpillar with parameters
(n; r; s2, s3, . . . , sr−1) and H is a graph of m vertices,
then P (T )= n − 1 and M(T) = 2n − r and
P (T ∧ H ) = P (T ) + nP (H ) + M(T)m
= n − 1 + nP (H ) + (2n − r)m.
Proof. First, it is clear that P (T ) |E(T)| = n − 1.
Next, suppose g is an M-optimal interval-labeling of T . Suppose u is the vertex for which g(u) has the smallest left endpoint, and v is the vertex other than u for which g(v) has the largest right endpoint. Let Q be the u–v path in T , and assume it has q vertices. Then, w∈V (Q)g(w) is an interval in the real line that
includes g(z) for any vertex z∈ V (T ). Consequently,
Mg(w) 1 for each w ∈ V (Q) and Mg(z) 2 for
each z∈ V (T ) − V (Q). As q r, we have M(T)
q+ 2(n − q) = 2n − q 2n − r.
On the other hand, define an interval-labeling g∗by
g∗(z)= [4i, 4i + 5], if z = xi, 1 i r; 4i+ 2 +2(s2j i+1), 4i+ 2 + 2j+1 2(si+1) , if z= yij, 1 < i < r, 1 j si.
It is clear that g∗ is an interval-labeling of T with
Ig∗(T)= n − 1 and Mg∗(T)= 2n − r. Consequently,
g∗ is both I -optimal and M-optimal, and P (T )=
n− 1 and M(T) = 2n − r. The theorem then follows
from Theorem 4. ✷
We next consider the case when G is the complete graph Kn.
Theorem 7. If Kn is a complete graph of n vertices
and H is a graph of m vertices, then P (Kn)=
n(n− 1)/2 and M(Kn)= (n + 1)2/4 and
P (Kn∧ H ) = P (Kn)+ nP (H ) + M(Kn)m = n(n − 1)/2 + nP (H )
+(n+ 1)2/4m.
Proof. First, it is clear that P (Kn) |E(Kn)| =
n(n− 1)/2.
Next, suppose g is an M-optimal interval-labeling of Kn. We recursively define u1, u2, . . . , un/2 and
v1, v2, . . . , vn/2 as follows. Having defined uj and
vj for all j < i, let ui be the vertex (other than the
above vertices) for which g(ui) has the smallest left
endpoint, and vi is the vertex (other than the above
vertices and ui) for which g(vi) has the largest right
endpoint. Repeat this process until all vertices of Kn
are scanned. It is the clear that for each 1 i
n/2 we have that g(ui)∪ g(vi) is an interval in the
real line that includes g(uj) and g(vj) for all j i.
Mg(vi) i for 1 i n/2. Then Mg(Kn) 1 +
2+ · · · + n/2 + 1 + 2 + · · · + n/2 = (n + 1)2/4.
On the other hand, suppose V (Kn)= {x1, x2,
. . . , xn}. Define g∗by g∗(xi)= [i, i +n] for 1 i n.
It is clear that g∗ is an interval-labeling of Kn with
Ig∗(Kn)= n(n − 1)/2 and Mg∗(Kn)= (n + 1)2/4.
Consequently, g∗ is both I -optimal and M-optimal, and P (Kn)= n(n − 1)/2 and M(Kn)= (n + 1)2/4.
The theorem then follows from Theorem 4. ✷ Finally, we consider the case when G is the cycle Cn.
Theorem 8. If Cn is a cycle of n vertices and H
is a graph of m vertices, then P (Cn)= 2n − 3 and
M(Cn)= 2n − 2 and
P (Cn∧ H ) = P (Cn)+ nP (H ) + M(Cn)m = 2n − 3 + nP (H ) + (2n − 2)m.
Proof. We first prove that P (Cn) 2n − 3 by
induc-tion on n. The case of n= 3 is clear as P (C3) |E(C3)| = 3. Suppose P (Cn) 2n − 3 for n 3.
Choose an I -optimal interval-labeling g of Cn+1. Let
x be the vertex whose g(x) has a smallest right
end-point. Then its two neighbors y and z both contain this right endpoint. Consequence g(y)∩ g(z) = ∅. Hence the restriction gof g on V (Cn+1)− {x} is an
interval-labeling of Cn. Then P (Cn+1)= Ig(Cn+1)
2+ Ig(Cn) 2 + P (Cn) 2 + 2n− 3 = 2(n+ 1) − 3.
Therefore, P (Cn) 2n − 3 for all n 3.
Next, suppose g is an M-optimal interval-labeling of Cn. Let u be the vertex for which g(u) has the
smallest left endpoint, and v be the vertex other than u for which g(v) has the largest right endpoint. Then Cn
is the union of two internally vertex-disjoint u–v paths
Q1and Q2. It is the case that g(x)⊆
y∈V (Q3−i)g(y)
for each vertex x∈ V (Qi), i= 1, 2. Consequently,
Mg(x) 2 for all vertices in Cn except Mg(u) 1
and Mg(v) 1. Then Mg(Cn) 2(n − 2) + 1 + 1 =
2n− 2.
On the other hand, suppose V (Cn)= {x1, x2,
. . . , xn}. Define g∗ by g∗(xi)= [2i, 2i + 3] for 1
i n − 1 and g∗(xn)= [3, 2n + 2]. It is clear that g∗
is an interval-labeling of Cn with Ig∗(Cn)= 2n − 3
and Mg∗(Cn)= 2n − 2. Consequently, g∗ is both
I -optimal and M-optimal, and P (Cn)= 2n − 3 and
M(Cn)= 2n − 2. The theorem then follows from
Theorem 4. ✷
Acknowledgements
The authors thank the referees for many construc-tive suggestions.
References
[1] A. Billionnet, On interval graphs and matrix profiles, RAIRO Rech. Opér. 20 (1986) 245–256.
[2] P.Z. Chinn, J. Chvatalova, A.K. Dewdney, N.E. Gibbs, The bandwidth problem for graphs and matrices—a survey, J. Graph Theory 6 (1982) 223–254.
[3] P.Z. Chinn, Y. Lin, J. Yuan, The bandwidth of the corona of two graphs, Congr. Numer. 91 (1992) 141–152.
[4] J.F. Fink, M.S. Jacobson, L.F. Kinch, J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287–293.
[5] F.V. Fomin, P.A. Golovach, Graph searching and interval completion, SIAM J. Discrete Math. 13 (2000) 454–464. [6] R. Frucht, F. Harary, On the coronas of two graphs,
Aequa-tiones Math. 4 (1970) 322–324.
[7] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, CA, 1979.
[8] N.E. Gibbs, W.G. Poole Jr., P.K. Stockmeyer, An algorithm for reducing the bandwidth and profile of a sparse matrix, SIAM J. Numer. Anal. 13 (1976) 235–251.
[9] Y. Guan, K. Williams, Profile minimization problem on trian-gulated triangles, Comput. Sci. Dept. Tech. Report, TR/98-02, Western Michigan University, Kalamazoo, MI, 1998. [10] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of
Domination in Graphs, Marcel Dekker, New York, 1998. [11] J. Jeffs, Effects of a local change on the bandwidth of a graph,
Congr. Numer. 89 (1992) 45–53.
[12] I.P. King, An automatic reordering scheme for simultaneous equations derived from network systems, Internat. J. Numer. Methods Engrg. 2 (1970) 523–533.
[13] B.U. Koo, B.C. Lee, An efficient profile reduction algorithm based on the frontal ordering scheme and the graph theory, Comput. Structures 44 (6) (1992) 1339–1347.
[14] D. Kuo, The profile minimization problem in graphs, Master Thesis, Dept. Applied Math., National Chiao Tung Univ., Hsinchu, Taiwan, June 1991.
[15] D. Kuo, G.J. Chang, The profile minimization problem in trees, SIAM J. Comput. 23 (1) (1994) 71–81.
[16] D. Kuo, J.-H. Yan, The profile on n-cubes, Preprint, 2000. [17] Y.-L. Lai, Bandwidth, Edgesum and profile of graphs, Ph.D.
Thesis, Department of Computer Science, Western Michigan University, Kalamazoo, MI, 1997.
[18] Y.-L. Lai, Exact profile values of some graph compositions, Taiwanese J. Math. 6 (1) (2002) 127–134.
[19] Y.-L. Lai, K. Williams, A survey of solved problems and applications on bandwidth, edgesum and profile of graphs, J. Graph Theory 31 (1999) 75–94.
[20] Y. Lin, J. Yuan, Profile minimization problem for matrices and graphs, Acta Math. Appl. Sinica, English-Series, Yingyong Shuxue-Xuebas 10 (1) (1994) 107–112.
[21] Y. Lin, J. Yuan, Minimum profile of grid networks, Systems Sci. Math. Sci. 7 (1) (1994) 56–66.
[22] X.K. Liu, Y. Li, Yan The incidence coloring of corona graphs, J. Xuzhou Norm. Univ. Nat. Sci. Ed. 19 (2) (2001) 16–18. [23] J.C. Luo, Algorithms for reducing the bandwidth and profile of
a sparse matrix, Comput. Structures 44 (3) (1992) 535–548. [24] J. Mai, Profiles of some condensable graphs, J. System Sci.
Math. Sci. 16 (1996) 141–148.
[25] W. Ning, L.Z. Zhang, The incidence coloring of corona graphs, J. Lanzhou Univ. Nat. Sci. 37 (3) (2001) 10–13.
[26] W. Ning, W. Zhang, F. Liu, The incidence coloring of corona graphs, J. Inn. Mong. Norm. Univ. 29 (2) (2000) 93–96. [27] C. Payan, N.H. Xuong, Domination-balanced graphs, J. Graph
Theory 6 (1982) 23–32.
[28] W.F. Smyth, Algorithms for the reduction of matrix bandwidth and profile, J. Comput. Appl. Math. 12&13 (1985) 551–561. [29] R.A. Snay, Reducing the profile of sparse symmetric matrices,
Bull. Geod. 50 (1976) 341–352.
[30] M. Wiegers, B. Monien, Bandwidth and profile minimization, Lecture Notes in Comput. Sci. 344 (1988) 378–392. [31] K. Williams, On the minimum sum of the corona of two graphs,
