Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

Transcription

1 Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs Andreas Brandstädt, Arne Leitert, Dieter Rautenbach University of Rostock University of Ulm

2 The Problem

3 Domination D V with N[d] = V d D 3 / 22

4 Domination D V with N[d] = V d D 3 / 22

5 Domination D V with N[d] = V d D 3 / 22

6 Efficient Domination Exactly one d D for each v V v V :! d D : v N[d] Exact cover of the closed neighbourhoods 4 / 22

7 Efficient Domination Exactly one d D for each v V v V :! d D : v N[d] Exact cover of the closed neighbourhoods 4 / 22

8 Efficient Domination Exactly one d D for each v V v V :! d D : v N[d] Exact cover of the closed neighbourhoods 4 / 22

9 Efficient Domination packing and covering problem 5 / 22

10 Efficient Domination packing and covering problem dominating 5 / 22

11 Efficient Domination packing and covering problem dominating efficient 5 / 22

12 Efficient Domination packing and covering problem dominating ED efficient 5 / 22

13 Efficient Domination Does not always exist 6 / 22

14 Efficient Domination Does not always exist 6 / 22

15 Efficient Domination Does not always exist 6 / 22

16 Efficient Domination Does not always exist 6 / 22

17 Efficient Edge Domination the same with edges 7 / 22

18 Efficient Edge Domination the same with edges 7 / 22

19 Efficient Edge Domination the same with edges 7 / 22

20 Other Names Efficient Domination Independent Perfect Domination Efficient Edge Domination Dominating Induced Matching 8 / 22

21 Existing Results ED co-comparability linear Chang et al 1995 planar bipartite, chordal bipartite NP-c. Lu, Tang 2002 chordal NP-c. Yenn, Lee 1996 EED P 7 -free linear ISAAC 2011 planar bipartite NP-c. Lu, Ko, Tang 2002 chordal linear Lu, Ko, Tang 2002 and more... 9 / 22

22 Our Solution

23 What is known By Definition The following are equivalent for any D: (i) D is EED in G (ii) D is ED in L(G) (iii) D is dominating set in L(G) and independent set in L(G) 2 11 / 22

24 What is known ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 11 / 22

25 What is known ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 11 / 22

26 What is known ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 11 / 22

27 Our Approach Vertex weight function ω : V N with ω(v) = N[v] 12 / 22

28 Our Approach Vertex weight function ω : V N with ω(v) = N[v] Theorem The following are equivalent for any D V : (i) D is ED in G (ii) D is min. w. dominating set in G with ω(d) = V (iii) D is max. w. independent set in G 2 with ω(d) = V 12 / 22

29 Our Approach Vertex weight function ω : V N with ω(v) = N[v] Theorem The following are equivalent for any D V : (i) D is ED in G (ii) D is min. w. dominating set in G with ω(d) = V (iii) D is max. w. independent set in G 2 with ω(d) = V (i) (iii) was also found by Martin Milanič. 12 / 22

30 Our Approach ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 13 / 22

31 Our Approach ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 13 / 22

32 Our Approach ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 13 / 22

33 Our Approach ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 13 / 22

34 Our Approach ω( ) G L( ) L(G) 2 L(G) 2 eed w. dom ed dom + ind w. ind w. ind 2 E(E)D 13 / 22

35 This includes for EED min. w. (perfect, independent) edge domination min. w. dominating matching max. w. induced matching 14 / 22

36 This includes for EED min. w. (perfect, independent) edge domination min. w. dominating matching max. w. induced matching... for ED min. w. perfect domination min. w. independent domination 14 / 22

37 Results dually chordal ED: linear using independent set for G 2 EED: linear G has EED dually chordal chordal This includes strongly chordal graphs. 15 / 22

38 Results AT-free ED: polynomial using independent set for G 2 (G 2 is AT-free) EED: polynomial using independent set for L(G) 2 (L(G) 2 is AT-free) 15 / 22

39 Results interval bigraphs ED: polynomial using domination for G 15 / 22

40 Results interval-filament EED: polynomial using independent set for L(G) 2 (L(G) 2 is interval-filament) 15 / 22

41 Results weakly chordal EED: polynomial using independent set for L(G) 2 (L(G) 2 is weakly chordal) 15 / 22

42 Results ED EED dually chordal linear linear AT-free polynomial polynomial interval bigraphs polynomial interval-filament polynomial weakly chordal polynomial 15 / 22

43 A View on Hypergraphs

44 Hypergraphs Hypergraph: H = (V, E) with E (V ) \ 17 / 22

45 Hypergraphs Hypergraph: H = (V, E) with E (V ) \ 17 / 22

46 Hypergraphs Hypergraph: H = (V, E) with E (V ) \ 17 / 22

47 Hypergraphs Hypergraph: H = (V, E) with E (V ) \ incidence graph of H I(H) 17 / 22

48 Duality Hypergraph H = (, ) I(H) 18 / 22

49 Duality Hypergraph H = (, ) Dual Hypergraph H = (, ) I(H) = I(H ) 18 / 22

50 Line / 2-Section graph I(H) 19 / 22

51 Line / 2-Section graph I(H) 2 19 / 22

52 Line / 2-Section graph line graph L(H) = I(H) 2 [E] L(H) 19 / 22

53 Line / 2-Section graph line graph L(H) = I(H) 2 [E] 2-Section graph 2Sec(H) = I(H) 2 [V ] 2Sec(H) 19 / 22

54 Efficient (Edge) Domination Definition on Hypergraphs D is ED in H D is ED in 2Sec(H). D is EED in H D is ED in L(H). 20 / 22

55 Efficient (Edge) Domination Definition on Hypergraphs D is ED in H D is ED in 2Sec(H). D is EED in H D is ED in L(H). Theorem H has an ED H has an EED. 20 / 22

56 Results on hypergraphs ED EED α-acyclic NP-complete polynomial hypertree polynomial NP-complete 21 / 22

57 Thank you for your attention!