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Introduction Exact results Bounds Conclusion Lower Bounds for the Power Domination Problem Daniela Ferrero Department of Mathematics Texas State University San Marcos, TX Great Plains Combinatorics Conference 2016 University of Kansas

  1. Introduction Exact results Bounds Conclusion Definitions Definition (Power dominating set) G = ( V , E ) graph, S ⊆ V power dominating set iff M i ( S ) = V for some i ∈ N . Definition (Power domination number) G = ( V , E ) graph, γ P ( G ) = min {| S | : S ⊆ V , ∃ i ∈ N : M i ( S ) = V } . Daniela Ferrero Lower Bounds for the Power Domination Problem

  2. Introduction Exact results Bounds Conclusion Definitions Definition (Power dominating set) G = ( V , E ) graph, S ⊆ V power dominating set iff M i ( S ) = V for some i ∈ N . Definition (Power domination number) G = ( V , E ) graph, γ P ( G ) = min {| S | : S ⊆ V , ∃ i ∈ N : M i ( S ) = V } . Definition (Power domination problem) G = ( V , E ) graph, find S ⊆ V such that γ P ( G ) = S and M i = V for some i ∈ N . Daniela Ferrero Lower Bounds for the Power Domination Problem

  3. Introduction Exact results Bounds Conclusion Basic Results This graph theory problem was introduces by Haynes et al. in 2002. Daniela Ferrero Lower Bounds for the Power Domination Problem

  4. Introduction Exact results Bounds Conclusion Basic Results This graph theory problem was introduces by Haynes et al. in 2002. Definition (Domination number) G = ( V , E ) graph, γ ( G ) = min {| S | : N [ S ] = V } = min {| S | : M 1 ( S ) = V } . Daniela Ferrero Lower Bounds for the Power Domination Problem

  5. Introduction Exact results Bounds Conclusion Basic Results This graph theory problem was introduces by Haynes et al. in 2002. Definition (Domination number) G = ( V , E ) graph, γ ( G ) = min {| S | : N [ S ] = V } = min {| S | : M 1 ( S ) = V } . Theorem (Haynes et al. 2002) For every graph G, 1 ≤ γ P ( G ) ≤ γ ( G ) . Daniela Ferrero Lower Bounds for the Power Domination Problem

  6. Introduction Exact results Bounds Conclusion Basic Results This graph theory problem was introduces by Haynes et al. in 2002. Definition (Domination number) G = ( V , E ) graph, γ ( G ) = min {| S | : N [ S ] = V } = min {| S | : M 1 ( S ) = V } . Theorem (Haynes et al. 2002) For every graph G, 1 ≤ γ P ( G ) ≤ γ ( G ) . Haynes et al. also characterized the extremal graphs G such that γ P ( G ) = 1 and those graphs G such that γ P ( G ) = γ ( G ). Daniela Ferrero Lower Bounds for the Power Domination Problem

  7. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Daniela Ferrero Lower Bounds for the Power Domination Problem

  8. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Daniela Ferrero Lower Bounds for the Power Domination Problem

  9. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Algorithmic results Daniela Ferrero Lower Bounds for the Power Domination Problem

  10. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Algorithmic results Exact results for families of graphs Daniela Ferrero Lower Bounds for the Power Domination Problem

  11. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Algorithmic results Exact results for families of graphs Effect of some graph operations Daniela Ferrero Lower Bounds for the Power Domination Problem

  12. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Algorithmic results Exact results for families of graphs Effect of some graph operations Upper bounds and lower bounds Daniela Ferrero Lower Bounds for the Power Domination Problem

  13. Introduction Exact results Bounds Conclusion Overview Theorem (Haynes et al. 2002) The power domination problem is NP-complete. Simulations Algorithmic results Exact results for families of graphs Effect of some graph operations Upper bounds and lower bounds Daniela Ferrero Lower Bounds for the Power Domination Problem

  14. Introduction Exact results Bounds Conclusion Rectangular grids Definition (Cartesian Product) G = ( V G , E G ) and H = ( V H , E H ) graphs, the Cartesian product G � H has V ( G � H ) = V G × V H and E ( G � H ) = { ( g , h )( g ′ , h ′ ) : g = g ′ , hh ′ ∈ E H or h = h ′ , gg ′ ∈ E G } Definition (Rectangular grid) The rectangular n × m grid is P n � P m where P n and P m are paths of order n and m respectively. Daniela Ferrero Lower Bounds for the Power Domination Problem

  15. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  16. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  17. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  18. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  19. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  20. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  21. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  22. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  23. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  24. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  25. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  26. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  27. Introduction Exact results Bounds Conclusion Rectangular grids Daniela Ferrero Lower Bounds for the Power Domination Problem

  28. Introduction Exact results Bounds Conclusion Rectangular grids We can extend the above construction for any n ≥ m ≥ 1 and prove: � � m +1 � if m ≡ 4 mod 8 4 γ P ( P n � P m ) ≤ � m � otherwise 4 Daniela Ferrero Lower Bounds for the Power Domination Problem

  29. Introduction Exact results Bounds Conclusion Rectangular grids We can extend the above construction for any n ≥ m ≥ 1 and prove: � � m +1 � if m ≡ 4 mod 8 4 γ P ( P n � P m ) ≤ � m � otherwise 4 The following result requires more work: Theorem (Dorfling & Henning 2006) � � m +1 � if m ≡ 4 mod 8 If n ≥ m ≥ 1 , γ P ( P n � P m ) = 4 � m � otherwise 4 Daniela Ferrero Lower Bounds for the Power Domination Problem

  30. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  31. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  32. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  33. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  34. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  35. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  36. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  37. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  38. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  39. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  40. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  41. Introduction Exact results Bounds Conclusion Cylinders Daniela Ferrero Lower Bounds for the Power Domination Problem

  42. Introduction Exact results Bounds Conclusion Cylinders We can extend the above construction for any n ≥ 1 and m ≥ 3 and prove: �� m + 1 � n + 1 � �� γ P ( P n � C m ) ≤ min , 4 2 Daniela Ferrero Lower Bounds for the Power Domination Problem

  43. Introduction Exact results Bounds Conclusion Cylinders We can extend the above construction for any n ≥ 1 and m ≥ 3 and prove: �� m + 1 � n + 1 � �� γ P ( P n � C m ) ≤ min , 4 2 The following result requires more work: Theorem (Barrera & DF 2011) �� m +1 � n +1 � �� If n ≥ 1 and m ≥ 3 , γ P ( P n � C m ) = min , 4 2 Daniela Ferrero Lower Bounds for the Power Domination Problem

  44. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  45. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  46. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  47. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  48. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  49. Introduction Exact results Bounds Conclusion Tori Daniela Ferrero Lower Bounds for the Power Domination Problem

  50. Introduction Exact results Bounds Conclusion Tori We can extend the above construction for any m ≥ n ≥ 3 and prove: � � n +1 � if n ≡ 2 mod 4 γ P ( C n � C m ) ≤ 2 � n � otherwise 2 Daniela Ferrero Lower Bounds for the Power Domination Problem

  51. Introduction Exact results Bounds Conclusion Tori We can extend the above construction for any m ≥ n ≥ 3 and prove: � � n +1 � if n ≡ 2 mod 4 γ P ( C n � C m ) ≤ 2 � n � otherwise 2 The following result requires more work: Theorem (Barrera & DF 2011) If n ≥ m ≥ 3 , � � n +1 � if n ≡ 2 mod 4 2 γ P ( C n � C m ) = � n � otherwise 2 Daniela Ferrero Lower Bounds for the Power Domination Problem

  52. Introduction Exact results Bounds Conclusion Zero Forcing Consider a black/white coloring of the vertices of a graph G . Daniela Ferrero Lower Bounds for the Power Domination Problem

  53. Introduction Exact results Bounds Conclusion Zero Forcing Consider a black/white coloring of the vertices of a graph G . Apply the following color changing rule, until its application does not change the color of any vertex: Daniela Ferrero Lower Bounds for the Power Domination Problem

  54. Introduction Exact results Bounds Conclusion Zero Forcing Consider a black/white coloring of the vertices of a graph G . Apply the following color changing rule, until its application does not change the color of any vertex: If a white vertex is the only white neighbor of a black vertex, then change its color to black. Daniela Ferrero Lower Bounds for the Power Domination Problem

  55. Introduction Exact results Bounds Conclusion Zero Forcing Consider a black/white coloring of the vertices of a graph G . Apply the following color changing rule, until its application does not change the color of any vertex: If a white vertex is the only white neighbor of a black vertex, then change its color to black. If at the end of the process all vertices are black, the initial set of black vertices is called a zero-forcing set . Daniela Ferrero Lower Bounds for the Power Domination Problem

  56. Introduction Exact results Bounds Conclusion Zero Forcing Consider a black/white coloring of the vertices of a graph G . Apply the following color changing rule, until its application does not change the color of any vertex: If a white vertex is the only white neighbor of a black vertex, then change its color to black. If at the end of the process all vertices are black, the initial set of black vertices is called a zero-forcing set . The zero-forcing number of G , denoted as Z ( G ), is the minimum cardinality of a zero-forcing set in G . Daniela Ferrero Lower Bounds for the Power Domination Problem

  57. Introduction Exact results Bounds Conclusion Zero Forcing & Power Domination Rules for power domination: Domination rule: A vertex power dominates itself and its neighbors. Propagation rule: If a power dominated vertex v has exactly one non-power dominated neighbor u , then v also power dominates u . Rule for zero forcing: If a white vertex is the only white neighbor of a black vertex, then change its color to black. Daniela Ferrero Lower Bounds for the Power Domination Problem

  58. Introduction Exact results Bounds Conclusion Zero Forcing & Power Domination Rules for power domination: Domination rule: A vertex power dominates itself and its neighbors. Propagation rule: If a power dominated vertex v has exactly one non-power dominated neighbor u , then v also power dominates u . Rule for zero forcing: If a white vertex is the only white neighbor of a black vertex, then change its color to black. Power domination = domination + zero forcing Daniela Ferrero Lower Bounds for the Power Domination Problem

  59. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Observation: If S is a power dominating set in a graph G then N [ S ] is a zero forcing set in G . If G has maximum degree ∆, | N [ S ] | ≤ | S | (∆ + 1) and if γ P ( G ) = | S | : Z ( G ) ≤ γ P ( G )(∆ + 1) so � Z ( G ) � ≤ γ P ( G ) ∆ + 1 Daniela Ferrero Lower Bounds for the Power Domination Problem

  60. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆ . Then, � Z ( G ) � ≤ γ P ( G ) . ∆ Daniela Ferrero Lower Bounds for the Power Domination Problem

  61. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆ . Then, � Z ( G ) � ≤ γ P ( G ) . ∆ ∪ s ∈ S N [ s ] − v s where v s is an arbitrary vertex Sketch of the proof: in N ( s ), is also a zero-forcing set. s 𝑤 $ Daniela Ferrero Lower Bounds for the Power Domination Problem

  62. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) If { s , t } is a power dominating set, N [ s ] ∪ N [ t ] is a zero forcing set but ( N [ s ] − { v s } ) ∪ ( N [ t ] − { v t } ) is not. 𝑤 $ s t 𝑤 " Daniela Ferrero Lower Bounds for the Power Domination Problem

  63. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) However, ( N [ s ] ∪ N [ t ]) − { v s } is still a zero forcing set. | ( N [ s ] ∪ N [ t ]) − { v s }| = | N [ s ] | + | N [ t ] − { v t }| − 1 ≤ (∆ + 1) + ∆ − 1 and we still have a zero forcing set of cardinality at most 2∆. 𝑤 $ s t 𝑤 " Daniela Ferrero Lower Bounds for the Power Domination Problem

  64. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Theorem (Benson et al. 2015) � � Z ( G ) Let G be a graph of maximum degree ∆ . Then, ≤ γ P ( G ) . ∆ Daniela Ferrero Lower Bounds for the Power Domination Problem

  65. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Theorem (Benson et al. 2015) � � Z ( G ) Let G be a graph of maximum degree ∆ . Then, ≤ γ P ( G ) . ∆ If A is the adjacency matrix of G and M = M ( A ) is its nullity, M ≤ Z ( G ) (Barioli et al 2008) Daniela Ferrero Lower Bounds for the Power Domination Problem

  66. Introduction Exact results Bounds Conclusion Z ( G ) and γ P ( G ) Theorem (Benson et al. 2015) � � Z ( G ) Let G be a graph of maximum degree ∆ . Then, ≤ γ P ( G ) . ∆ If A is the adjacency matrix of G and M = M ( A ) is its nullity, M ≤ Z ( G ) (Barioli et al 2008) Corollary (Benson at al. 2015) Let G be a graph of maximum degree ∆ and adjacency matrix A � � M with nullity M = M ( A ) . Then, ≤ γ P ( G ) . ∆ Daniela Ferrero Lower Bounds for the Power Domination Problem

  67. Introduction Exact results Bounds Conclusion Rectangular Grids Let M n × m be the n × m rectangular grid, 1 ≤ m ≤ n , Z ( M n × m ) ≤ γ P ( M n × m ) 4 Daniela Ferrero Lower Bounds for the Power Domination Problem

  68. Introduction Exact results Bounds Conclusion Rectangular Grids Let M n × m be the n × m rectangular grid, 1 ≤ m ≤ n , Z ( M n × m ) ≤ γ P ( M n × m ) 4 Z ( M n × m ) = min { n , m } Daniela Ferrero Lower Bounds for the Power Domination Problem

  69. Introduction Exact results Bounds Conclusion Rectangular Grids Let M n × m be the n × m rectangular grid, 1 ≤ m ≤ n , Z ( M n × m ) ≤ γ P ( M n × m ) 4 Z ( M n × m ) = min { n , m } � m � ≤ γ P ( M n × m ) 4 Daniela Ferrero Lower Bounds for the Power Domination Problem

  70. Introduction Exact results Bounds Conclusion Rectangular Grids Let M n × m be the n × m rectangular grid, 1 ≤ m ≤ n , Z ( M n × m ) ≤ γ P ( M n × m ) 4 Z ( M n × m ) = min { n , m } � m � ≤ γ P ( M n × m ) 4 A simple construction shows γ P ( M n × m ) ≤ ⌈ m +1 4 ⌉ so, Daniela Ferrero Lower Bounds for the Power Domination Problem

  71. Introduction Exact results Bounds Conclusion Rectangular Grids Let M n × m be the n × m rectangular grid, 1 ≤ m ≤ n , Z ( M n × m ) ≤ γ P ( M n × m ) 4 Z ( M n × m ) = min { n , m } � m � ≤ γ P ( M n × m ) 4 A simple construction shows γ P ( M n × m ) ≤ ⌈ m +1 4 ⌉ so, � � m +1 � if m ≡ 4 mod 8 4 γ P ( M n × m ) = � m � otherwise 4 Daniela Ferrero Lower Bounds for the Power Domination Problem

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