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 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. 3 / 11

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 . 4 / 11

Dominating Set for Trees

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 . 6 / 11

Dominating Set for Trees 1 Pick an arbitrary vertex s . 2 Compute a BFS-order σ = � s = v 1 , v 2 , . . . , v n � . 3 For i := n downto 1 If v i is not dominated Then 4 Add the parent u of v to the set D and mark all neighbours 5 of u as dominated. (We consider s as its own parent.) Runs in O ( | V | ) time. 7 / 11

Variants of Dominating Sets

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 . 9 / 11

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? 10 / 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 G 2 with ω ( D ) = | V | . Leitert 2013; Milaniˇ c 2013 G 2 = ( V , E 2 ) with E 2 = { uv | u � = v , d G ( u , v ) ≤ 2 } 11 / 11