Peng and Chang [170] gave an elegant algorithm for the domatic partition prob-lem in strongly chordal graphs.
Their algorithm uses a primal-dual approach. Suppose [v1, v2, . . . , vn] is a strong elimination ordering of G = (V, E) with the minimum degree δ(G).
Choose a vertex x of degree δ(G). As any dominating set Di in a domatic partition of G contains at least one vertex vi in N [x] and two distinct Di have different corresponding vi, it is easy to see the following inequality.
Weak Duality Inequality: d(G) ≤ δ(G) + 1.
Their algorithm maintains δ(G) + 1 disjoint sets. Initially, these sets are empty. The algorithm scans the vertices in the reverse order of the strong elimination ordering. A vertex is included in a set when it is scanned. When the algorithm terminates, these δ(G) + 1 sets are dominating sets.
A vertex v is completely dominated if v is dominated by all of these δ(G) + 1 dominating sets.
Algorithm DomaticSC. Determine a domatic partition of a strongly chordal graph G of size δ(G) + 1.
Input. A strongly chordal graph G = (V, E) with a strong elimination ordering [v1, v2, . . ., vn].
Output. A partition of V into δ(G) + 1 disjoint dominating sets of G.
Method.
Si← ∅ for 1 ≤ i ≤ δ(G) + 1;
for i = n to 1 step −1 do
find the largest k ∼ i such that vk is not completely dominated;
let Sℓbe a set that does not dominate vk; Sℓ← {vi} ∪ Sℓ;
if no such set exists then include vito an arbitrary Sℓ; end do.
Before proving the correctness of the algorithm, two lemmas are needed.
Lemma 6.3 Assume Sℓ⊆ {vi+1, vi+2, . . . , vn} and k ∼ i, where 1 ≤ i ≤ n. If Sℓ does not dominate vk, then Sℓdoes not dominate vj for all j ≤ k with j ∼ i.
Proof. Suppose to the contrary that Sℓ has a vertex vp dominating vj, i.e., i < p and p ∼ j. Then i < p, j ≤ k, i ∼ j, i ∼ k and p ∼ j imply p ∼ k by (SEO), which contradicts that Sℓ does not dominate vk.
Let r(v) = |{x ∈ N [v]: x is not in any of the δ(G) + 1 sets}| and ndom(v) be the number of sets that do not dominate v during the execution of Algorithm DomaticSC.
Lemma 6.4 Algorithm DomaticSC maintains the following invariant:
r(vj) ≥ ndom(vj) for all j ∈ {1, 2, . . . , n}.
Proof. The lemma is proved by induction. Initially,
r(vj) = deg(vj) + 1 ≥ δ(G) + 1 = ndom(vj)
for all vj ∈ V . During iteration i, only values of r(vj) and ndom(vj), where j ∼ i, may be altered when vi is included in a set Sℓ. Notice that the algorithm determines the largest index k ∼ i such that vk is not completely dominated. It then finds a set Sℓ that does not dominate vk (Sℓ is chosen arbitrarily when vk
does not exist).
For any j ∼ i with j ≤ k, by Lemma 6.3, vj was not dominated by Sℓ. Therefore, r(vj) and ndom(vj) are decremented by one after vi is included in Sℓ.
On the other hand, for any j ∼ i with j > k (or non-existence of such vk), by the choice of the vertex vk in the algorithm, vertex vj is completely dominated, i.e., ndom(vj) = 0. Thus the invariant is maintained.
Theorem 6.5 Algorithm DomaticSC partitions the vertex set of a strongly chordal graph G = (V, E) into d(G) = δ(G) + 1 disjoint dominating sets in linear time provided that a strong elimination ordering is given.
Proof. Upon termination of the algorithm, r(vj) = 0 for all j ∈ {1, 2, . . . , n}.
According to Lemma 6.4, ndom(vj) = 0 for all vj in V . That is, these δ(G) + 1 sets are dominating sets of G. The strong duality equality
d(G) = δ(G) + 1 then follows from the weak duality inequality.
To implement that algorithm efficiently, each vertex vi is associated with a variable ndom(i) and an array L(i) of size δ(G)+1. Initially, ndom(i) = δ(G)+1 and the values of entries in Li are all zero. Thus, for each vertex it takes O(di) time to test ndom(i) to determine vk, where diis the degree of vi. It then takes O(δ(G) + 1) time to decide which set vishould go. Finally, for each vj∈ N [vi], it takes O(1) time to update ndom(j) and Lj. Therefore, the algorithm takes
O Xn
i=1(di+ δ(G) + 1)
!
= O(|V | + |E|) time.
7 Permutation graphs
Given a permutation π = (π(1), π(2), . . . , π(n)) on the set In = {1, 2, . . . , n}, the permutation graph of π is the graph G(π) = (In, E(π)) with
E(π) = {jk : (j − k)(π−1(j) − π−1(k)) < 0}.
Note that π−1(j) is the position of j in the permutation π. Figure 13 illustrates a permutation and its corresponding permutation graph. If a line between each integer i and its position in π is drown, then n lines are created, each with an associated integer. In this way, two vertices j and k are adjacent in G(π) if and only if their corresponding lines cross. That is, G(π) is the intersection graph of these n lines. Notice that an independent set in G(π) corresponds to an increasing subsequence of π, and a clique in G(π) corresponds to a decreasing subsequence of π.
Permutation graphs were first introduced by Pnueli, Lempel and Even in [172, 89]. Since that time quite a few polynomial time algorithms have been
3 6 4 1 8 7 5 8 7 6 5 4 3 2 1
2HH
HHHHH
Q QQ
QQQ
i
i i i
i i
i i
2 1 3
4 6 7
5 8
) π = (
G(π)
Figure 13: A permutation and its corresponding permutation graph.
constructed on permutation graphs. For example, Atallah, Manacher and Ur-rutia [9], Brandst¨adt and Kratsch [32], Farber and Keil [96] and Tsai and Hsu [192] have constructed polynomial domination and independent domination al-gorithms, Brandst¨adt and Kratsch [32] and Colbourn and Stewart [80] have constructed polynomial connected domination algorithms, and Brandst¨adt and Kratsch [32] and Corneil and Stewart [80] have constructed polynomial total domination algorithms.
This section presents a simple O(n2)-time algorithm, due to Brandst¨adt and Kratsch [33] for finding a minimum weighted independent dominating set in a permutation graph. Assume that the defining permutation π of the permutation graph is given as part of the input. Spinrad [187] has shown that π can be constructed in O(n2) time, given the graph G.
This algorithm takes the advantage of the observation that a set is an in-dependent dominating set if and only if it is a maximal inin-dependent set. Since maximal independent sets in permutation graphs correspond to maximal in-creasing subsequences in π, all that is necessary is to search for such a sequence in π of minimum weight. In particular, it determines, for every j, 1 ≤ j < n, the minimum weight γi(j, w) of an independent dominating set in the subsequence π(1), π(2), . . . , π(j), which contains π(j) as the rightmost element. Let w(j) denote the weight of vertex j.
Algorithm WIndDomPer. Solve the weighted independent domination prob-lem for permutation graphs.
Input. A permutation graph G with its corresponding permutation π on the set {1, 2, . . . , n} and vertex weights w(1), w(2), . . . , w(n) of real numbers.
Output. The weighted independent domination number γi(G, w) of G.
Method.
for j = 1 to n do p(j) ← 0;
γi(j) = w(π(j));
end do;
for k = j − 1 to 1 step −1 do
if (π(j) > π(k) and p(j) = 0) then γi(j) ← w(π(j)) + γi(k);
p(j) ← π(k);
end if;
else if π(j) > π(k) > p(j) > 0 then γi(j) ← min{γi(j), w(π(j)) + γi(k)};
p(j) ← π(k);
end if end do;
m ← p(n);
γi(G, w) ← γi(n);
for j = n − 1 to 1 step −1 do if π(j) > m then
γi(G, w) ← min{γi(G, w), γi(j)};
m ← π(j);
end if.
The algorithm is illustrated by the permutation graph G(π) in Figure 13, where π = (2 3 6 4 1 8 7 5) and all weights are equal to 1: γi(j) = (1 2 3 3 1 2 2 2), and thus the minimum size of an independent dominating set is 2, for example the set {1, 5}.
8 Cocomparability graphs
A graph G = (V, E) is a comparability graph if G has an orientation H = (V, F ) such that xy, yz ∈ F imply xz ∈ F . In other words, G has a comparability ordering which is an ordering [v1, v2, . . . , vn] of V satisfying
i < j < k and vivj, vjvk∈ E imply vivk∈ E. (CO) A graph G = (V, E) is a cocomparability graph if its complement G is a com-parability graph, or equivalently, G has a cocomcom-parability ordering which is an ordering [v1, v2, . . . , vn] of V satisfying
i < j < k and vivk ∈ E imply vivj ∈ E or vjvk∈ E. (CCO) There is an O(n2.376)-time recognition algorithm for comparability graphs and thus for cocomparability graphs [187]. This has been improved by Mc-Connell and Spinrad who gave an O(n + m)-time algorithm constructing an orientation of any given graph G such that the orientation is a transitive orien-tation of G if and only if G has a transitive orienorien-tation [161]. Unfortunately, the best algorithm for testing whether the orientation is indeed transitive has running time O(n2.376).
The class of cocomparability graphs is a well studied superclass of the classes of interval graphs, permutation graphs and trapezoid graphs. Domination prob-lems on cocomparability graphs were considered for the first time by Kratsch
and Stewart [149]. They obtained polynomial time algorithms for the domi-nation/total domination/connected domination and the weighted independent domination problems in cocomparability graphs. These algorithms are designed by dynamic programming using cocomparability orderings. Breu and Kirk-patrick [35] (see [4]) improved this by giving O(nm2)-time algorithms for the domination and the total domination problems and an O(n2.376)-time algorithm for the weighted independent domination problem in cocomparability graphs.
On the other hand, the weighted domination, total domination and con-nected domination problems are NP-complete in cocomparability graphs [51].
Also, the problem “Given a cocomparability graph G, does G have a dominating clique?” is NP-complete [149].
An O(n3)-time algorithm computing a minimum cardinality connected dom-inating set of a connected cocomparability graph has been given in [149]. The following is the algorithm for this problem given by Breu and Kirkpatrick [35]
(see also [4]).
Let [v1, v2, . . . , vn] be a cocomparability ordering of a cocomparability graph G = (V, E). For vertices u, w ∈ V , write u < w if u appears before w in the ordering, i.e., u = vi and w = vj implies i < j. For i ≤ j the set {vk : i ≤ k ≤ j} is denoted by [vi, vj]. Then vivj ∈ E implies that every vertex vk with i < k < j is adjacent to vi or to vj, thus {vi, vj} dominates [vi, vj]. This can be generalized as follows: let S ⊆ V where G[S] is connected. Then S dominates [min(S), max(S)] where min(S) (respectively, max(S)) is the vertex of S with the smallest (respectively, largest) index in the ordering.
The following theorem and lemma are given in [149].
Theorem 8.1 Any connected cocomparability graph G has a minimum con-nected dominating set S such that the induced subgraph G[S] is a chordless path p1, p2, . . . , pk.
Lemma 8.2 Suppose S ⊆ V is a minimum connected dominating set of a co-comparability graph G = (V, E) with a coco-comparability ordering [v1, v2, . . . , vn].
If G[S] is a chordless path p1, p2, . . . , pk, then every vertex x < min(S) is dom-inated by {p1, p2} and every vertex y > max(S) is dominated by {pk−1, pk}.
The following approach enables an elegant way of locating a chordless path of minimum size that dominates the cocomparability graph. A source vertex is a vertex vi such that vkvi∈ E for all k < i and a sink vertex is a vertex vj such that vjvk ∈ E for all k > j. Then [v1, v2, . . . , vn] is a canonical cocomparability ordering if [v1, v2, . . . , vr], 1 ≤ r < n, are the source vertices and vs, vs+1, . . . , vn, 1 < s ≤ n, are the sink vertices. Note that every cocomparability graph G has a canonical cocomparability ordering. Furthermore, given any cocomparability ordering, a canonical one can be computed in time O(n + m).
From now on, assume that [v1, v2, . . . , vn] is a canonical cocomparability ordering. Since the source vertices of G form a clique, any source vertex vi
dominates [v1, vi]. Analogously, since the sink vertices of G form a clique, any sink vertex vjdominates [vj, vn]. Therefore the vertex set of every path between a source vertex and a sink vertex is dominating.
The following theorem given in [35] enlights the key property.
Theorem 8.3 Every connected cocomparability graph G = (V, E) satisfying γc(G) ≥ 3 has a minimum connected dominating set which is the vertex set of a shortest path between a source and a sink vertex of G.
Proof. Let [v1, v2, . . . , vn] be a canonical cocomparability ordering of G. Ac-cording to Theorem 8.1, there is a minimum connected dominating set S of G such that G[S] is a chordless path P : p1, p2, . . . , pk, k ≥ 3. Construct below a chordless path P′′between a source vertex and a sink vertex of G, that has the same number of vertices as the path P .
Let p1 = vi and p2 = vj. First observe that p2 = vj cannot be a source vertex, otherwise N [p2] ⊇ [v1, vj] implying that {p2, p3, . . . , pk} is also a con-nected dominating set of G, a contradiction. If p1 is a source vertex then P starts at a source vertex. In this case, proceed with the path P′ = P (possibly) rearranging pk−1, pk.
Suppose p1= vi is not a source vertex. Then there is a source vertex u of G with up16∈ E. Since [v1, v2, . . . , vn] is a canonical cocomparability ordering and since p1 and p2 are not source vertices, u < p1and u < p2. Since vivj∈ E and by Lemma 8.2, {vi, vj} dominates [1, max{vi, vj}]. Consequently up2∈ E.
Consider the set S′ = {u, p2, . . . , pk}. Since {u, p2} dominates [v1, vj], S′ is a dominating set. Since P : p1, p2, . . . , pk is a chordless path, t ≥ 3 implies ptu 6∈ E. Thus S′ induces the chordless path P′: u, p2, . . . , pk.
Similarly, starting from P′the vertex pkcan be replaced, if necessary. Vertex pk−1 is not a sink vertex. If pk is a sink vertex then S′′ = S′ and P′′ = P′. Otherwise, replace pk by a sink vertex v satisfying vpk 6∈ E to obtain S′′ and P′′.
G[S′′] induces a chordless path between a sink and a source vertex. The vertex set of any path between a sink and a source vertex is a dominating set.
By construction S′′ is a minimum connected dominating set. Consequently S′′
is the vertex set of a shortest path between a source vertex and a sink vertex of G.
According to Theorem 8.3, when γc(G) ≥ 3, computing a minimum con-nected dominating set of a concon-nected cocomparability graph G reduces to com-puting a shortest path between a source and a sink vertex of G.
Algorithm ConDomCC. Solve the connected domination problem in cocom-parabiliy graphs.
Input: A connected cocomparability graph G = (V, E) and a canonical cocom-parability ordering [v1, v2, . . . , vn] of G.
Output: A minimum connected dominating set of G.
Methods.
1. Check whether G has a minimum connected dominating set D of size at most 2. If so, output D and stop.
2. Construct a new graph G′ by adding two new vertices s and t to G such that s is adjacent exactly to the source vertices of G and t is adjacent exactly to the sink vertices of G.
3. Compute a shortest path P : s, p1, p2, . . . , pk, t between s and t in G′ by the breadth-first-search.
4. Output {p1, p2, . . . , pk}.
The correctness of Algorithm ConDomCC follows immediately from Theo-rem 8.3. The “almost linear” running time of the algorithm follows from the well-known fact that breadth-first-search is a linear-time procedure.
Theorem 8.4 For any connected cocomparability graph G = (V, E) with a canonical cocomparability ordering [v1, v2, . . . , vn], Algorithm ConDomCC out-puts a minimum connected dominating set of G in O(nm) time. In fact, all parts of the algorithm except checking for a connected dominating set of size two can be done in time O(n + m).
It is clearly unsatisfactory that the straightforward test for a connected dom-inating set of size two dominates the overall running time. The crux is that there are even permutation graphs for which each minimum connected dominating set of size two contains neither a source nor a sink vertex (see [129, 143]) It seems that minimum dominating sets of this type cannot be found by a shortest path approach. It is an open question whether Step 1 of Algorithm ConDomCC can be implemented in a more efficient way.
Notice that the O(n)-time algorithms computing a minimum connected dom-inating set for permutation graphs [129] and trapezoid graphs [143] both rely on Theorem 8.3.
Corneil, Olariu and Stewart have done a lot of research on asteroidal triple-free graphs, usually called AT-triple-free graphs [74, 77]. They are defined as those graphs not containing an asteroidal triple, i.e., a set of three vertices such that between any two of the vertices there is a path avoiding the neighborhood of the third.
AT-Free graphs form a superclass of the cocomparability graphs. They are a “large class of graphs” with nice structural properties and some of them are related to domination. One of the major theorems on the structure of AT-free graphs states that every connected AT-free graph has a dominating pair, i.e., a pair of vertices u, v such that the vertex set of each path between u and v is a dominating set.
An O(n + m) algorithm computing a dominating pair for a given connected AT-free graph has been presented in [77]. This can be used to obtain an O(n+m) algorithm computing a dominating path for connected AT-free graphs (see also [75]). An O(n3) algorithm computing a minimum connected dominating set for
connected AT-free graphs is given in [11]. An O(n + m) algorithm computing a minimum connected dominating set in connected AT-free graphs with diameter greater than three is given in [77].