Weighted connected k-domination and weighted
k-dominatingclique in distance-hereditary graphs
Hong-Gwa Yeh
a, Gerard J. Chang
b; ∗aDepartment of Mathematics, National Central University, Chungli 320, Taiwan bDepartment of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road,
Hsinchu 30050, Taiwan Accepted April 2000
Abstract
A graph is distance-hereditary if the distance between any two vertices in a connected in-duced subgraph is the same as in the original graph. This paper presents e-cient algorithms for solvingthe weighted connected k-domination and the weighted k-dominatingclique problems in distance-hereditary graphs. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Distance-hereditary graph; Connected k-domination; k-dominatingclique; Algorithm
1. Introduction
In a graph G, the distance dG(x; y) between two vertices x and y is the minimum length of an x–y path. A graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. Distance-hereditary graphs were introduced by Howorka [14], who gave the 8rst characterization of these graphs. Further characterizations and optimization problems in these graphs were then extensively studied in the literature, see the references. The purpose of this paper is to present e-cient algorithms for solving the weighted connected k-domination and the weighted k-dominatingclique problems in distance-hereditary graphs.
To consider these two problems on a graph G, every vertex v of G is associated with a nonnegative integer k(v) (the domination restriction function) and a nonnegative weight w(v) (the weight function). The weight w(S) of a vertex set S is the sum of the weights of its elements. A vertex subset D is a connected k-dominating set of G if D induces a connected subgraph and every vertex v of G is k-dominated by some
Supported in part by the National Science Council under grant NSC87-2115-M009-007.
∗Correspondingauthor. Tel.: +886-3-573-1945; fax: +886-3-542-2682. E-mail address: [email protected] (G.J. Chang).
0304-3975/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
vertex x in D, i.e., dG(v; x)6k(v). The weighted connected k-domination problem is to determine the weighted connected k-domination number c(G; k; w), which is the minimum weight of a connected k-dominatingset of G.
In the above de8nitions, if in addition the set D is a clique then we have the weighted k-dominating clique problem and the weighted k-dominating clique number clique(G; k; w). For the case when all w(v) are 1, we have unweighted versions of the problems. For the case when all k(v) are 1, we have domination for k-domination.
Domination and its variations, originates from location problems in operations re-search, have been extensively studied in the literature, see the references. In general, determining c(G) and clique(G) and NP-complete [12, 15]. On the other hand, there are e-cient algorithms for the connected k-domination and the k-dominatingclique problems in strongly chordal graphs [5, 11] and dually chordal graphs [9, 1], which include strongly chordal graphs.
As for distance-hereditary graphs, D’Atri and Moscarini [8] 8rst gave an O(|V ||E|)-time algorithm for the connected domination problem; Yeh and Chang [21] developed a linear-time algorithm for the weighted connected domination problem; BrandstIadt and Dragan [2] presented a linear-time algorithm for the connected k-domination problem. Also, Dragan [10] gave a linear-time algorithm for the k-dominatingclique problem in distance-hereditary graphs. Combining their techniques, this paper presents e-cient algorithms for the weighted connected k-domination and the weighted k-dominating clique problems in distance-hereditary graphs. For technical reasons, we de8ne
c(G; k; w; u) = min{w(D) : D is a connected k-dominatingset with u ∈ D} and
clique(G; k; w; u) = min{w(D) : D is a k-dominatingclique with u ∈ D}:
For convenience, we call a subset D of V a c(k; u)-set of G if D is a connected k-dominatingset of G such that u ∈ D.
2. Preliminaries for distance-hereditary graphs
Suppose A and B are two vertex sets in a graph G = (V; E). G[A] denotes the sub-graph of G induced by A. The neighborhood NA(B) of B in A is the set of vertices in A that are adjacent to some vertex in B. The closed neighborhood NA[B] of B in A is NA(B) ∪ B. For simplicity, NA(v); NA[v]; N(B); and N[B] stand for NA({v}); NA[{v}]; NV(B), and NV[B], respectively.
The hanging hu of a connected graph G = (V; E) at a vertex u ∈ V is the collection of sets (called levels) L0(u); L1(u); : : : ; Lt(u) (or L0; L1; : : : ; Lt if there is no ambigu-ity), where t = maxv∈V dG(u; v) and Li(u) = {v ∈ V : dG(u; v) = i} for 06i6t. For any 16i6t and any vertex v ∈ Li, let N(v) = N(v) ∩ Li−1. A vertex v ∈ Li with 16i6t has a minimal neighborhood in Li−1 if N(x) is not a proper subset of N(v) for any x ∈ Li.
The followingproperties of distance-hereditary graphs are useful in this paper. Theorem 1 (D’Atri and Moscarini [8]). A connected graph G = (V; E) is distance-hereditary if and only if for every hanging hu= (L0; L1; : : : ; Lt) of G and every pair of vertices x and y in Li with 16i6t in the same component of G[V −Li−1]; we have N(x) = N(y).
Theorem 2 (Hammer and MaKray [13, Fact 3.4]). Suppose hu= (L0; L1; : : : ; Lt) is a hanging of a connected distance-hereditary graph at u. For each 16i6t; there exists a vertex v ∈ Li such that v has a minimal neighborhood in Li−1. In addition; if v satis9es the above condition then NV −N(v)(x) = NV −N(v)(y) for every pair of vertices
x and y in N(v).
3. The algorithm
We now establish theorems that are basis of the e-cient algorithms for the weighted connected k-domination and the weighted k-dominatingclique problems in distance-hereditary graphs.
Theorem 3. Suppose hu= (L0; L1; : : : ; Lt) is a hanging of a connected distance-hereditary graph G = (V; E) at u. Let k be a domination restriction function and D be a c(k; u)-set of G. Then for each x ∈ Li with i¿1 and k(x)¿1; there exists a vertex y ∈ D ∩ Lj for some j¡i which k-dominates x.
Proof. We 8rst note that for each vertex z ∈ Lp, a shortest u–z path has the form P : u = v0; v1; : : : ; vp= z, where vr∈ Lr for 16r6p. Since D is a c(k; u)-set of G, by the de8nition, there exists a vertex y∈ D and an x–y path P
1: x = x0; x1; : : : ; xd= y such that d6k(x). We may assume that y∈ D ∩ Lj and y is the only vertex of P1
that is in D. If j¡i, then we may choose y = y to k-dominate x as desired. Next,
we consider the case of j¿i. Since G[D] is connected, there is a shortest u–y path
P2: u = y0; y1; : : : ; yj= y in G usingvertices only in D, where yr∈ Lr for 06r6j.
Let s be the smallest index such that Ls∩ P1= ∅, say xr∈ Ls∩ P1. Since 16s6i6j, the vertices x ∈ Li and xr; ys∈ Ls are connected in G[V − Ls−1] through the x–ys walk:
x = x0; x1; : : : ; xr; xr+1; : : : ; xd= y= yj; yj−1; : : : ; ys:
For the case of xr= y, we have s = i = j. By Theorem 1, N(x) = N(ys) and so x is adjacent to ys−1. We may then choose y = ys−1∈ D∩Ls−1with s−1¡i to k-dominate x as desired. For the case of xr= y, we have r6d−1. Again, N(xr) = N(ys) by Theorem 1 and so xr is adjacent to ys−1. Thus,
dG(x; ys−1)6dG(x; xr) + 16(d − 1) + 16k(x)
and so we may choose y = ys−1 to k-dominate x as desired. This completes the proof of the theorem.
Theorem 4. Suppose hu= (L0; L1; : : : ; Lt) is a hanging of a connected distance-heredit-ary graph G = (V; E) at u. Let k be a domination restriction function; w a nonnegative weight function; and x a vertex in Lt having a minimal neighborhood B = NLt−1(x).
Choose y ∈ B with w(y) = minb∈Bw(b) and z ∈ B with k(z) = minb∈Bk(b). Suppose G= G − x; the weight function w is the restriction of w on V (G); and k is de9ned
by k(v) = 0 if v = y with k(z) ¿ k(x) = 0; k(x) − 1 if v = y with k(z)¿k(x)¿1; k(v) otherwise:
Then the following statements hold:
(1) c(G; k; w; u) = c(G; k; w; u) + w(x) when k(x) = 0; and c(G; k; w; u) = c(G; k; w; u) otherwise.
(2) Suppose G has a k-dominating clique containing vertex u. If k(x) = 0; then
clique(G; k; w; u) = clique(G; k; w; u).
Proof. We only give the proof for (1). The proof for (2) is similar and thus omitted. Suppose D is a minimum weighted c(k; u)-set of G. Since G[D] is connected and u ∈ D, every vertex of D − {u} is adjacent to a vertex of D in a lower level. This, together with Theorem 3, implies that we may assume x =∈ D when k(x)¿1, and that D − {x} is a k-connected dominatingset of G. For the case of k(z)¿k(x) = 0; B
contains some vertex in D and so, by Theorem 2, we may assume that y ∈ D. For the case of k(z)¿k(x)¿1, by Theorem 3, x is within distance k(x) in G from a vertex
x∈ D in some level j¡t. When j = t − 1, we may assume x= y; and in any case,
y is within distance k(x) − 1 in G from x. Accordingto the de8nition, k is the
same as k except the above two cases; and so, by Theorem 3, D − {x} is a connected k-dominatingset of G. Therefore,
c(G; k; w; u)6 c(G; k; w; u)−w(x) when k(x) = 0, and c(G; k; w; u)6 c(G; k; w; u) otherwise.
On the other hand, suppose D is a connected k-dominatingset of G. For the
case of k(x) = 0, accordingto the de8nition of k, either k(y) = 0 or k(z) = k(z) = 0.
Hence, B contains a vertex of D. Then D ∪ {x} is a connected k-dominatingset of
G and so c(G; k; w; u)6 c(G; k; w; u) + w(x). For the case of k(x)¿1, according to the de8nition of k; B contains a vertex within distance k(x) − 1 in G from a
vertex x∈ D. Then x is within distance k(x) in G from x∈ D. Therefore, D is a
k-connected dominatingset of G and so c(G; k; w; u)6 c(G; k; w; u). The above inequalities imply the result in (1).
Based on Theorem 4, we have the following O(|V ||E|)-time algorithm for computing c(G; k; w) of a connected distance-hereditary graph G = (V; E). The algorithm can also be modi8ed to solve the weighted k-dominatingclique problem in the same time complexity.
begin
c(G; k; w) ← ∞; for i = 1 to n do begin
for j = 1 to n do Mk(uj) = k(uj); Mk(ui) ← 0;
p ← w(ui); = ∗ p stands for c(G; k; w; ui) ∗ =
determine the hanging hui= (L0; L1; : : : ; Lt) of G at ui;
for j = t downto 1 do begin
sort Lj= {x1; x2; : : : ; x‘} such that
|N(x p1)|6|N(xp2)|6 · · · 6|N(xp‘)|; for r = 1 to ‘ do begin let y; z ∈ B = N(x pr) such that
w(y) = minb∈Bw(b) and Mk(z) = minb∈B Mk(b); if Mk(z) ¿ Mk(xpr) = 0 then Mk(y) ← 0; if Mk(z)¿ Mk(xpr)¿1 then Mk(y) ← Mk(xpr) − 1; if Mk(xpr) = 0 then p ← p + w(xpr); end end c(G; k; w) ← min{ c(G; k; w); p} end end. References
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