Lower Bounds for the Power Domination Problem Daniela Ferrero - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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