SLIDE 1
Dominating Set
Colouring For a graph G = (V, E), a vertex set D is called dominating set if
- v∈D
N[v] = V. If v ∈ D and u ∈ N[v], we say that v dominates u.
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Domination Dominating Set Colouring For a graph G = ( V , E ) , a - - PowerPoint PPT Presentation
Domination Dominating Set Colouring For a graph G = ( V , E ) , a - - PowerPoint PPT Presentation
Domination Dominating Set Colouring For a graph G = ( V , E ) , a vertex set D is called dominating set if N [ v ] = V . v D If v D and u N [ v ] , we say that v dominates u . 2 / 11 Domonating Set Problem Dominating Set
SLIDE 2
SLIDE 3
Domonating Set Problem
Dominating Set Problem Given a graph G find a dominating set D such that |D| is minimal. Theorem There is (probably) no polynomial time algorithm to find a minimum dominating set. We can get a reasonable approximation in O(|E| log |V|) time.
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SLIDE 4
Approximation for Dominating Set
Algorithm
◮ Pick a vertex v such that the number of non-dominated vertices in
N[v] is maximal and add v to the set D.
◮ Repeat until each vertex is dominated.
Theorem If D∗ is a minimum dominating set, the algorithm computes a dominating set D such that |D∗| ≤ |D| ≤ (1 + ln |V|)|D∗|. Theorem There is (probably) no polynomial time algorithm which guaranties to find a dominating set D such that |D| ≤ c log |V||D∗| for some c > 0.
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SLIDE 5
Dominating Set for Trees
SLIDE 6
Dominating Set for Trees
Lemma Let v be a leaf in T with the parent u. There is a minimum dominating set D with u ∈ D. Proof.
◮ Assume that u /
∈ D. Then, v ∈ D.
◮ Because N[v] ⊆ N[u], D′ := D + u − v is a minimum dominating
set.
- Note. This only works, because v has only one neighbour. To make it
work for multiple neighbours, we would require that, for all w ∈ N[v], N[w] ⊆ N[u]. In this case, u is called am maximum neighbour of v.
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SLIDE 7
Dominating Set for Trees
1 Pick an arbitrary vertex s. 2 Compute a BFS-order σ = s = v1, v2, . . . , vn. 3 For i := n downto 1 4
If vi is not dominated Then
5
Add the parent u of v to the set D and mark all neighbours
- f u as dominated. (We consider s as its own parent.)
Runs in O(|V|) time.
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SLIDE 8
Variants of Dominating Sets
SLIDE 9
Variants of Dominating Sets
Independent Domination
◮ D is an independent set.
Connected Domination
◮ D induces a connected subgraph.
r-Domination
◮ r : V → N. For all v, there is a u ∈ D with d(u, v) ≤ r(v).
Total Domination
◮ For all v, N(v) ∩ D = ∅, i. e., a vertex v does not dominate itself.
Perfect Domination
◮ For all v /
∈ D, |N[v] ∩ D| = 1. Efficient Domination
◮ For all v, |N[v] ∩ D| = 1.
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SLIDE 10
Efficient Domination
Packing and covering problem
◮ Enough vertices to dominate everything. ◮ Few enough that they not overlap.
Not every graph contains a dominating set. Efficient Domination Problem
◮ Does G contain an efficient dominating set?
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SLIDE 11
Efficient Domination
Theorem If G has an efficient dominating set, then all efficient dominating sets have the same cardinality.
Bange et al. 1988
Consider the vertex weight function ω(v) =
- N[v]
- .
Theorem The following are equivalent for a graph G: (i) G has an efficient dominating set D (ii) D is a minimum weight dominating set in G with ω(D) = |V|. (iii) D is a maximum weight independent set in G2 with ω(D) = |V|.
Leitert 2013; Milaniˇ c 2013 G2 = (V, E2) with E2 = { uv | u = v, dG(u, v) ≤ 2 }
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