Weighted connected domination and Steiner trees in distance-hereditary graphs

DISCRETE

APPLIED

MATHEMATICS

ELSEVIER Discrete Applied Mathematics 87 (1998) 245-253

Weighted connected domination and Steiner trees in

distance-hereditary

graphs *

Hong-Gwa Yeh, Gerard J. Chang”

Depurtment of Applied Mathematics, Nationul Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan

Received 20 September 1994 received in revised form 3 March 1998; accepted 9 March 1998

Abstract

Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph. Both problems are MY-complete in general graphs. 0 1998 Elsevier Science B.V. All rights reserved.

Keywords:

Distance-hereditary graph; Connected domination; Steiner tree; Algorithm; Cograph

1. Introduction

The concept of domination can be used to model many location problems in op- erations research. In a graph G =

(V,E),

a

dominating set

is a subset

D

of vertices such that every vertex in

V - D

is adjacent to some vertex in

D.

A dominating set of G is

connected

if the subgraph

G[D]

induced by

D

is connected. The

connected

domination problem

is to find a minimum-sized connected dominating set of a graph. Suppose, moreover, that each vertex u in G is associated with a weight W(V) that is a real number. The

weighted connected domination problem

is to find a connected dominating set

D

such that

w(D) = COED w(v)

is as small as possible.

The concept of Steiner trees originally concerned points in Euclidean spaces, but it is also closely related to connected domination in graphs. Suppose T is a subset of vertices in a graph G =

(V, E).

The

Steiner tree problem

is to find a minimal subset S of

V - T

such that G[S U T] is connected. S and

T

are called the

Steiner set

and

target set,

respectively. We can also consider the

vertex-weighted version

of the Steiner tree problem, which was originally introduced by Segev [27]. The vertex weight of a

* Supported in part by the National Science Council under grant NSC84-2121-M009-023. * Corresponding author. E-mail: [email protected].

0166-218X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved.

246 H.-G. Yeh, G.J. Changi Discrete Applied Mathematics 87 (1998) 245-253

Steiner vertex can be interpreted as the cost of adding this vertex when forming the tree. Traditionally, the problem of finding the Steiner tree for a set of points in a graph has been studied for edge-weighted graphs (see [21]). However, Johnson [22, p. 445, line 91, pointed out that the edge-weighted Steiner tree problem is NP-complete for any classes that contains all complete graphs. In particular, the edge-weighted Steiner tree problem is _&Y-complete for the edge-weighted distance-hereditary graphs as these contain the edge-weighted complete graphs. So, we only consider the vertex-weighted Steiner tree problem for distance-hereditary graphs in this paper.

The connected domination and Steiner tree problems have the same complexity for many classes of graphs. For instance, they are both polynomially solvable for strongly chordal graphs [30], permutation graphs [7], cographs [9, 221, series-parallel graphs [12, 26, 29, 301, and distance-hereditary graphs [3, 131; and they are MY-complete for bipartite graphs [18, 251, split graphs [23, 301, chordal graphs [23, 301, and chordal bipartite graphs [24]. It is also known that the connected domination problem is poly- nomially solvable for k-trees (fixed k) [l] and ~-CUBS [l l] and MY-complete for ~-CUBS (k 22) [ 1 I]. The Steiner tree problem is polynomially solvable in homo- geneous graphs [ 141.

For many location problems, the corresponding domination problems may have differ- ent constraints or objective functions. Typical examples are r-domination and weighted versions. Results for these variant domination problems are relatively fewer than the usual version. Some well-known results of this kind are polynomial algorithms for the weighted domination and the weighted independent domination problems in strongly chordal graphs [ 171, the weighted perfect domination problem in co-comparability graphs [6], the r-domination problems in trees [28] and strongly chordal graphs [5], the connected r-domination problem in strongly chordal graphs [S] and distance-hereditary graphs [3]. The purpose of this paper is to present linear-time algorithms for the weighted connected domination problem with arbitrary weights and the vertex-weighted Steiner tree problem with non-negative weights in distance-hereditary graphs.

In the rest of this section, we give a brief survey of distance-hereditary graphs. A graph is distance-hereditary if every two vertices have the same distance in every connected-induced subgraph. Distance-hereditary graphs were introduced by Howorka [20]. The characterization and recognition of distance-hereditary graphs have been stud- ied in [2, 13, 15, 19, 201. Note that the class of distance-hereditary graphs is a subclass of all parity graphs [4] and a superclass of all cographs [8, lo].

Suppose A and B are two sets of vertices in a graph G = (V,E). The neighborhood No of B in A is the set of vertices in A that are adjacent to some vertex in B. The closed neighborhood &[B] of B in A is NA[B]UB. For simplicity, NA(u), NA[u], N(B),

and

NW stand for NA({D}), N~[{u}l, NO>,

and Nv[B], respectively. The distance dC(x, y) or d(x,

y)

between two vertices x and y in G is the minimum length of an x-y path in G. The hanging h, of a connected graph G = (V, E) at a vertex 11 E V is the collection of sets LO(U), L,(u), . . . , L,(u) (or Lo, Li,. . ,Lt if there is no ambiguity), where t = maxVEvdo(u,v) and Li(U) = {u E V : d~(u,v) = i} for O<i<t. For any

H.-G. Yeh, G.J. Changi Discrete Applied Mathematics 87 (1998) 245-253 247

v E U n Li with 1 <i < t has a minimal neighborhood in Li-1 with respect to U if N’(w) is not a proper subset of N’(v) for any w E U

n

Li. When U = V we omit the term U in the above definition.

Theorem 1

(Bandelt and Mulder [2], D’Atri and Moscarini [13], and Day et al. [15]).

For a connected graph G = (V, E) the following statements are equivalent: (1) G is a distance-hereditary graph.

(2) Every cycle of length at least jive in G has two crossing chords.

(3) For every hanging h, = (Lo,Ll , . . . , L,) of G and every pair of vertices x, y E L; (1 didt) that are in the same component of G[V-L,_I], we have N’(x) = N’(y).

Theorem 2

(Bandelt and Mulder [2]). Suppose h, = (Lo,L,,...,L,) is a hanging of

a connected distance-hereditary graph at u. For any two vertices x, y E Li with i3 1, N’(x) and N’(y) are either disjoint, or one of the two sets is contained in the other.

Theorem 3

(Fact 3.4 in Hammer and Maffray [19]). Suppose h, = (Lo,L,, . . . , L,) is

a hanging of a connected distance-hereditary graph at u. For each 1 <i< t, there exists a vertex v E Li such that v has a minimal neighborhood in Li-1. In addition, tf v satisfies the above condition then for every pair of vertices x and y in N’(v), we have NV-N+)(X) = NV-NJ(&Y).

2.

Weighted connected domination

This section presents a linear-time algorithm for finding a minimum weighted con- nected dominating set of a connected distance-hereditary graph G = (V,E) in which each vertex v has a weight w(v) that is a real number.

Lemma 4.

Suppose G = (V,E) is a connected graph with a weight function w on

V. Let V’ be the set of vertices v with w(u) < 0 and w’ be defined by w’(v) = m={w(v), O> f or all v E V. If D is a minimum w’-weighted connected dominating set of G, then D u V’ is a minimum w-weighted connected dominating set of G.

Proof.

First of all, since D is a connected dominating set of G, D U V’ is also. Next, suppose M is a minimum w-weighted connected dominating set of G. Since M is a connected dominating set of G and D is a minimum w’-weighted connected dominating set of G, w’(M)> w’(D), i.e., w(M - V’) = w’(M)>w’(D) = w(D - V’), and so

w(M)=w(M-V’)+w(MflV’)~w(D-V’)+w(V’)=w(DuV’). This completes the proof of the lemma. 0

Lemma 4 suggests that it suffices to consider the weighted connected domination problem with a non-negative weight function.

248 H.-G. Yeh, G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253

Lemma 5. Suppose h, = {Lo,Ll,.

. .,

L,} is a hanging of a connected distance-

hereditary graph at u. For any connected dominating set D and v E Li with 2 <i < t,

D n N’(v) # 0.

Proof.

Choose a vertex y in

D

that dominates v. Then y E

Li_1

U

Li

U

Li+l.

If y E

Li_1,

then y E

D n

N’(V). SO we may assume that y E

Li

U

Li+l.

Choose a vertex x E

Dfl(LoULl)

and an x-y path

P:

x = 211,212,. . . ) 0, = y

using vertices only in

D.

Let j be the smallest index such that {Vj, Vj+l, . . . , v,} C LiU

Li+l

U ’ ” U

Lt.

Then Uj E

Li, vi-1 E N’(vj),

and v and Uj are in the same component of G[V -

Li_l].

By Theorem 1 (3), N’(V) = N’(vj) and SO vi-1 E

DnN’(v).

In any case,

D r-7

N’(v) # 0. 0

Theorem 6. Suppose G = (V, E) is a connected distance-hereditary graph with a non-

negative weight function w on its vertices. Let h, = {LO,

LI,

.

. . ,

L,} be a hanging at a

vertex u of minimum weight. Consider the set d = {N’(v)

: v E

Li with 2 <

i

6 t and

v has a minimal neighborhood in Li_I}. For each N’(v) in d, choose one vertex v*

in N’(v) of minimum weight, and let D be the set of all such v*. Then D or D

U {u}

or some {v} with v E V is a minimum weighted connected dominating set of G.

Proof.

For any x E

Li

with 2

<i < t,

by Theorem 2, N’(x) includes some N’(u) in d. Thus we have Claim 1.

Claim 1. For any x E Li with 2 <i< t, x is adjacent to some vertex in Li_1 n D.

Claim 2. D

U {u}

is a connected dominating set of G.

Proof of Claim 2.

By Claim 1 and N[u] =

L1

U {u},

D

U {u} is a dominating set of G.

Also, by Claim 1, for any vertex x in

D

U {u} there exists an x-u path using vertices only in

D

U {u}, i.e.,

G[D

U {u}] is connected. ? ?

Suppose M is a minimum weighted connected dominating set of G. By Lemma 5, M n N’(v) # 0 for each N’(v) E d, say u** E M n N’(v). Note that any two sets in d are disjoint, so [Ml>]&’ =

IDI.

Case

1: ]M] = 1. The theorem is obvious in this case.

Case 2: ]MI > IDI.

In this case, there is at least one vertex x in M that is not a ii++. So

This together with Claim 2 proves that

D

U {u} is a minimum weighted connected dominating set of G.

H.-G. Yeh. G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253 249

Case 3: IMI = IDI 22. Since d contains pairwise disjoint sets, M = {u**: N’(u) E d}. so w(M) = c,** w(u**)> c,. w(v*) = w(D).

For any two vertices x* and y* in D, x** and y** are in M. Since G[M] is connected, there is an x**-y** path in G[M]:

**

x =v ** _{0 2} v, ** ) . . . )

v;* = y

**

.

For any 1 di<n, since v: and uf’ are both in N’(Vi) E d, by Theorem 3, Nv__NI(~,,) (UT) = NV-~+,)(u;*). But u,!:, EN Y N~(~~,)(uT*). So vFT1 E Nv__N/(~,)(u~) and v; E _ NV-,,,:,,_, ,(ui*_*, ). Also, that o;_i and vi*_r, are both in N’(Ui_ 1) E d implies that NV-N+_,)(u,?-,) = NV-N+_, )(~i+-*~). Then o,* E N v - ~t(~>,_,)(u,*_~). This proves that vi*-, is adjacent to VT for 1 <i d n and then

x* = l&v; ) . . . ) v; = y*

is an x*-y* path in G[D], i.e., G[D] is connected.

For any x in V, since A4 is a dominating set, x E N[u**] for some N’(v) E &. Note that v** and v* are both in N’(v). By Theorem 3, Nv_N/(~)(u**) = NY_~~(~)(u*). In the case of x $! N’(u), x E N[v**] implies x E N[u*], i.e., D dominates x. In the case of x E N’(o), Nv_,~/(~)(D*) = NY-NJ(~)(X). Since G[D] is connected and IDI >2, u* is adjacent to some y* E D - N’(u). Then x is also adjacent to y*, i.e., D dominates X. In any case, D is a dominating set. Therefore, D is a minimum weighted connected dominating set of G. 0

By Lemma 4 and Theorem 6, we can design an efficient algorithm for the weighted connected domination problem in distance-hereditary graphs. To implement the algo- rithm efficiently, we do not actually find the set d. Instead, we perform the following step for each 2 <i <t. Sort the vertices in Li such that

IN’(xi)I 6 IN’(xz)I < < (N’(xj)I.

We then process N’(xk) for k from 1 to j. At iteration k, if N’(xk)

n

D = 0, then

N’(xk) is in d and we choose a vertex of minimum weight to put it into D; otherwise, N'(xB ) +! d and we do nothing.

Algorithm WCD-dh.

Find a minimum weighted connected dominating set of a con-

nected distance-hereditary graph.

Input: A connected distance-hereditary graph G = (I’, E) and a weight w(u) of real number for each v E V.

Output: A minimum weighted connected dominating set D of graph G.

begin

D -

0;

let V’ = (0 E V : w(v) < O}; w(u) c 0 for each v E V’;

let u be a vertex of minimum weight in V;

250 H.-G. Yeh, G. J. Chung I Discrete Applied Mathematics 87 (1998) 245-253

for

i = 2

to

t

do

begin

let Li = {Xt,...,Xj};

SOI? Li such that JN’(Xi, )I 6 lN’(Xi2)/ 6 . . . < IN’(Xi,)l;

for

k = 1

to j do

end

if N’(XiJ n D = 8 then

D +-- D U {y} where y is a vertex of minimum weight in N’(xi,)

if

not (Lt CN[D] and G[D] is connected)

then

D - D U {u};

for v E V

that dominates V:

if w(v) <

w(D)

then

D - {v};

D-DUV’

end

Theorem 7. Algorithm WCD-dh gives a minimum weighted connected dominating set

of a connected distance-hereditary graph in linear time.

Proof.

The correctness of the algorithm follows from Lemma 4 and Theorem 6. For

each i, we can sort Li by using a bucket sort. So the algorithm is linear to ) VI + IEl. 0

3. Vertex-weighted Steiner tree

This section presents a linear-time algorithm for finding a minimum vertex-weighted Steiner tree with respect to a target set T C V in a connected distance-hereditary graph G = (V,E) with a non-negative weight w(v) for each v E V. In this section h, = {Lo, . . . , L,} denotes a hanging of G at a vertex u in the target set T. The key to our algorithm for the vertex-weighted Steiner tree problem is the following theorem. The theorem is similar to Theorem 6, but even simpler.

Theorem 8.

Suppose U = {x E V : x lies

on a shortest u-v path for some v in T}

and &J = {N’(x)

: x E U n

Li has

a minimal neighborhood in

Li-1

relative to U and

N’(x) n T =

0).

Then the set S formed by choosing a vertex x*

of

minimum weight

in each N’(x) E 99 is a minimum vertex-weighted Steiner set with respect to T in G.

Proof.

We first note that for each u-v path

P

with v E

Li, P

is a shortest path if and only if

P

is of the form

U=VO,V~ ,..., Vi=V,

where Vj E

Lj

for 0 <j < i. Therefore N’(x) C: U for each x E

U.

Consequently, S u T c

U.

For each x E SUT, x E U. Either N’(x) E 9# or N’(x)nT # 0 or N’(x) 2 N’(y) for some N’(y) E 29. Then there exists some vertex z E N’(x) rl (S u

T).

The same

H.-G. Yeh. G. J. Chang I Discrete Applied Mathematics 87 (1998) 245-253 251 argument can be applied repeatedly to show that there exists a shortest x-u path using vertices only in S U T. This proves that G[S U T] is connected.

Next, suppose M is a minimum vertex-weighted Steiner set with respect to T in G. For each N’(x) E ~8, by the definition of U, x lies on a shortest u-v path

u = VO,VI ,...) x = v I)..., vj = v

for some v in T, where each t.& E Lk. Since G[M LI T] is connected, there path

is a u-v

u = uo,uj )...) u, )...) us = v

in G[M U T], where ~~-1 E Li_1, ur E Li, and u,tl,...,u, are in Lk’s with k>i. By Theorem 1 (3), N’(x) = N’(u,). Therefore N’(x) contains u,_t E A4 U T. Since T n N’(x) = 0, A4 f? N’(x) # 8. Since any two sets in &J are disjoint and x* is a vertex

of minimum vertex-weight in N’(x), we conclude that w(M) > w(S). This proves that S is a minimum vertex-weighted Steiner set with respect to T in G. 0

Theorem 8 provides the basic idea for designing a good algorithm for the vertex- weighted Steiner tree problem. Similar to the implementation of WCD-dh, we do not actually find U and 2J. Instead, at any Li we sort IN’(x)/ for all x E S U T to find g.

Algorithm WST-dh.

Find a minimum vertex-weighted Steiner set of a distance-

hereditary graph with non-negative weights on its vertices.

Input: A connected distance-hereditary graph G = (V,E) with non-negative weight w(v) for each u E V and a subset T C V.

Output: A subset S C V - T of minimum weight such that G[S U T] is connected.

begin

s-

_0;

let u be a vertex of T;

determine the hanging h, = (LO, L1

,

. . . , L, ) of G at U;

for

i = t

to 2 step

-1

do

begin

let (Su T)nLi = {x,,...,x~};

if

p # 0

then

begin

sort x1,x2,...,

xp such that ~N’(.x,, )I < . . . < IN’(xj,,)j;

for

k = 1

to

p

do

if N’(Xjk)

has no vertices in S U T

then S - S U {y} where JJ is a vertex of minimum weight in N/(x], )

end

end

end

252 H.-G. Yeh, G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253

Theorem 9. Algorithm WST-dh solves the vertex-weighted Steiner tree problem for

a connected distance-hereditary graph with a non-negative weight function in linear

time.

Proof. Similar to the proof of Theorem 7.

? ?

The vertex-weighted Steiner tree problem with arbitrary real weights on its vertices

remains open.

Acknowledgements

The authors thank referees for many constructive suggestions on the revision of this

paper.

References

[l] S. Amborg, A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math. 23 (1989) I l-24.

[2] H.J. Bandelt, H.M. Mulder, Distance-hereditary graphs, J. Combin. Theory, Series B 41 (1986) 182-208.

[3] A. Brandstidt, F.F. Dragan, A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs, preprint, 1994.

[4] M. Burlet, J.P. Uhry, Parity graphs, Annals of Discrete Math. 21 (1984) 253-277.

[5] G.J. Chang, Labeling algorithms for domination problems on sun-free chordal graphs, Discrete Appl. Math. 22 (1989) 21-34.

[6] G.J. Chang, C. Pandu Rangan, Satyan R. Coorg, Weighted independent perfect domination on cocomparability graphs, Discrete Appl. Math. 63 (1995) 215-222.

[7] C.J. Colboum, L.K. Stewart, Permutation graphs: connected domination and Steiner trees, Discrete Math. 86 (1990) 145-164.

[8] D.G. Comeil, H. Lerchs, L. Stewart, Complement reducible graphs, Discrete Appl. Math. 3 (1981) 163-174.

[9] D.G. Comeil, Y. Perl, Clustering and domination in perfect graphs, Discrete Appl. Math. 9 (1984) 27-39.

[lo] D.G. Comeil, Y. Perl, L.K. Stewart, A linear recognition algorithm for cographs, SIAM J. Comput. 14 (1985) 926934.

[l l] D.G. Comeil, L.K. Stewart, Dominating sets in perfect graphs, Ann. Discrete Math. 86 (1990) 145-164. [12] G. Comuejols, J. Fonlupt, D. Naddef, The traveling salesman problem on a graph and some related

integer polyhedra, Math. Programming 33 (1985) l-27.

[13] A. D’Atri, M. Moscarini, Distance-hereditary graphs, Steiner trees, and connected domination, SIAM J. Comput. 17 (1988) 521-538.

[14] A. D’Atri, M. Moscarini, A. Sassano, The Steiner tree problem and homogeneous sets, Lecture Notes in Computer Science, vol. 324, Springer, Berlin, 1988, pp. 249-261.

[15] D.P. Day, O.R. Oellermann, H.C. Swart, Steiner distance-hereditary graphs, SIAM J. Discrete Math. 7 ( 1994) 437442.

[ 161 A.-H. Esfahanian, O.R. Oellermann, Distance-hereditary graphs and multidestination message-routing in multicomputers, JCMCC 13 (1993) 213-222.

[ 171 M. Farber, Domination, independent domination and duality in strongly chordal graphs, Discrete Appl. Math. 7 (1984) 115-130.

[18] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, New York, 1979.

H.-G. Yeh, G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253 253 [19] P.H. Hammer, F. Maffray, Completely separable graphs, Discrete Appl. Math. 27 (1990) 85-99.

[20] E. Howorka, A characterization of distance-hereditary graphs, Quart. J. Math. Oxford Ser. 2, 28 (1977) 4 17420.

[21] F.K. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem, North-Holland, Amsterdam, 1992. [22] D.S. Johnson, The NP-completeness column: an ongoing guide, J. Algorithms 6 (1985) 434451. [23] R. Laskar, J. Pfaff, Domination and irredundance in split graphs, Technical Report 430, Clemson

University, 1983.

[24] H. Miiller, A. Brandstidt, The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs, Theoret. Comput. Sci. 53 (1987) 2577265.

[25] J. Pfaff, R. Laskar, ST. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, 1983.

[26] R.L. Rardin, R.G. Parker, M.B. Richey, A polynomial algorithm for a class of Steiner tree problems on graphs, ISE Report J-82-5, Georgia Institute of Technology, 1982.

[27] A. Segev, The node-weighted Steiner tree problem, Networks 17 (1987) I-17. [28] P.J. Slater, R-Domination in graphs, J. Assoc. Comp. Mach. 23 (1976) 446450.

[29] J.A. Wald, C.J. Colboum, Steiner tree, partial 2-trees, and minimum IFI networks, Networks 13 (1983) 159-167.

[30] K. White, M. Farber, W.R. Pulleyblank, Steiner trees, connected domination and strongly chordal graphs, Networks 15 (1985) 109-124.