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].
9 Distance-hereditary graphs
This section presents a linear-time algorithm given by Yeh and Chang [202]
for finding a minimum weighted connected 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. According to Lemma 2.1, assume that the vertex weights are nonnegative.
Lemma 9.4 Suppose hu= {L0, L1, . . . , Lt} is a hanging of a connected distance-hereditary graph at u. For any connected dominating set D and v ∈ Li with 2 ≤ i ≤ t, D ∩ N′(v) 6= ∅.
Proof. Choose a vertex y in D that dominates v. Then y ∈ Li−1∪ Li∪ Li+1. If y ∈ Li−1, then y ∈ D ∩ N′(v). So, assume that y ∈ Li∪ Li+1. Choose a vertex x ∈ D ∩ (L0∪ L1) and an x-y path
P : x = v1, v2, . . . , vm= y
using vertices only in D. Let j be the smallest index such that {vj, vj+1, . . . , vm}
⊆ Li∪ Li+1∪ . . . ∪ Lt. Then vj ∈ Li, vj−1 ∈ N′(vj), and v and vj are in the same component of G − Li−1. By Theorem 9.1, N′(v) = N′(vj) and so vj−1∈ D ∩ N′(v). In any case, D ∩ N′(v) 6= ∅.
Theorem 9.5 Suppose G = (V, E) is a connected distance-hereditary graph with a non-negative weight function w on its vertices. Let hu = {L0, L1,. . . , Lt} be a hanging at a vertex u of minimum weight. Consider the set A = {N′(v): v ∈ Li with 2 ≤ i ≤ t and v has a minimal neighborhood in Li−1}.
For each N′(v) in A, 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} or some {v} with v ∈ V is a minimum weighted connected dominating set of G.
Proof. For any x ∈ Li with 2 ≤ i ≤ t, by Theorem 9.2, N′(x) includes some N′(v) in A. This gives Claim 1.
Claim 1. For any x ∈ Li with 2 ≤ i ≤ t, x ∈ N [Li−1∩ D].
Claim 2. D ∪ {u} is a connected dominating set of G.
Proof of Claim 2. By Claim 1 and N [u] = L1∪ {u}, D ∪ {u} is a dominating set of G. Also, by Claim 1, for any vertex x in D ∪ {u} there exists an x-u path using vertices only in D ∪ {u}, i.e., G[D ∪ {u}] is connected. 2
Suppose M is a minimum weighted connected dominating set of G. Accord-ing to Lemma 9.4, M ∩ N′(v) 6= ∅ for each N′(v) ∈ A, say v∗∗ ∈ M ∩ N′(v).
Since any two sets in A are disjoint, |M | ≥ |A| = |D|.
Case 1. |M | = 1. The theorem is obvious in this case.
Case 2. |M | > |D|. In this case, there is at least one vertex x in M that is not a v∗∗. Then
w(M ) ≥X
v∗∗w(v∗∗) + w(x) ≥X
v∗w(v∗) + w(u) = w(D ∪ {u}).
This together with Claim 2 gives that D ∪{u} is a minimum weighted connected dominating set of G.
Case 3. |M | = |D| ≥ 2. Since A contains pairwise disjoint sets, M = {v∗∗: N′(v) ∈ A}. Then w(M ) =P
v∗∗w(v∗∗) ≥P
v∗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∗∗= v0∗∗, v∗∗1 , . . . , v∗∗n = y∗∗.
For any 1 ≤ i ≤ n, since v∗i and vi∗∗ are both in N′(vi) ∈ A, by Theorem 9.3, NV\N′(vi)(v∗i) = NV\N′(vi)(v∗∗i ). But vi−1∗∗ ∈ NV\N′(vi)(vi∗∗). Therefore vi−1∗∗ ∈ NV\N′(vi)(v∗i) and vi∗∈ NV\N′(vi−1)(vi−1∗∗ ). Also, that vi−1∗ and vi−1∗∗ are both in N′(vi−1) ∈ A implies that NV\N′(vi−1)(vi−1∗ ) = NV\N′(vi−1)(vi−1∗∗ ). Then vi∗ ∈ NV\N′(vi−1)(v∗i−1). This proves that vi−1∗ is adjacent to vi∗ for 1 ≤ i ≤ n and then
x∗= v0∗, v∗1, . . . , vn∗ = y∗ is an x∗-y∗ path in G[D], i.e., G[D] is connected.
For any x in V , since M is a dominating set, x ∈ N [v∗∗] for some N′(v) ∈ A. Note that v∗∗ and v∗ are both in N′(v). According to Theorem 9.3, NV\N′(v)(v∗∗) = NV\N′(v)(v∗). In the case of x 6∈ N′(v), x ∈ N [v∗∗] implies x ∈ N [v∗], i.e., D dominates x. In the case of x ∈ N′(v), NV\N′(v)(v∗) = NV\N′(v)(x). Since G[D] is connected and |D| ≥ 2, v∗ is adjacent to some y∗ ∈ D \ N′(v). 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.
Lemma 2.1 and Theorem 9.5 together give an efficient algorithm for the weighted connected domination problem in distance-hereditary graphs as fol-lows. To implement the algorithm efficiently, the set A is not actually found.
Instead, the following step is performed for each 2 ≤ i ≤ t. Sort the vertices in Li such that
|N′(x1)| ≤ |N′(x2)| ≤ . . . ≤ |N′(xj)|.
Then process N′(xk) for k from 1 to j. At iteration k, if N′(xk) ∩ D = ∅, then N′(xk) is in A and choose a vertex of minimum weight to put it into D;
otherwise N′(xk) 6∈ A and do nothing.
Algorithm WConDomDH. Find a minimum weighted connected dominating set of a connected distance-hereditary graph.
Input: A connected distance-hereditary graph G = (V, E) and a weight w(v) of real number for each v ∈ V .
Output: A minimum weighted connected dominating set D of graph G.
Method.
D ← ∅;
let V′= {v ∈ V : w(v) < 0};
w(v) ← 0 for each v ∈ V′;
let u be a vertex of minimum weight in V ;
determine the hanging hu= (L0, L1, . . . , Lt) of G at u;
for i = 2 to t do
let Li= {x1, x2, . . . , xj};
sort Li such that |N′(xi1)| ≤ |N′(xi2)| ≤ . . . ≤ |N′(xij)|;
for k = 1 to j do
if N′(xik) ∩ D = ∅ then D ← D ∪ {y} where y is a vertex of minimum weight in N′(xik);
end do;
if not (L1⊆ N [D] and G[D] is connected) then D ← D ∪ {u};
for v ∈ V that dominates V do if w(v) < w(D) then D ← {v};
D ← D ∪ V′.
Theorem 9.6 Algorithm WConDomDH 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 2.1 and Theorem 9.5. For each i, sort Li by using a bucket sort. Then the algorithm runs in O(|V | + |E|) time.
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