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

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 Lawrence, KS

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Co-authors

Roberto Barrera

Texas A&M University

Daniela Ferrero

Texas State University

Leslie Hogben

Iowa State University

Michael Young

Iowa State University

Franklin H. Kenter

Rice University

Katie Benson

Westminster College

Veronika Furst

• Ft. Lewis College

Violeta Vasilevska

Utah Valley University

Mary Flagg

University of St. Thomas

Brian Wissman

University of Hawaii

“Power domination and zero forcing” Submitted, 2015 “Propagation time and lower bounds for the power domination number” Submitted, 2015 “Power domination in cylinders, tori and generalized Petersen graphs” Networks, 2011

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Electric Power Network

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Monitoring

Electric power networks must be continuously monitored to prevent blackouts and power surges.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Monitoring

Electric power networks must be continuously monitored to prevent blackouts and power surges. Most broadly used method: placing phase measurement units (PMUs) at selected network locations.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Monitoring

Electric power networks must be continuously monitored to prevent blackouts and power surges. Most broadly used method: placing phase measurement units (PMUs) at selected network locations. PMUs measure magnitude and phase angle of the electric wave at the locations where they are placed.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Monitoring Process

PMU PMU PMU

• - - - - - - -

GPS satellite Power Control Center

If PMU readings are synchronized via GPS it is possible to use Kirchhoff’s laws to calculate the electric waves at any network location.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Problem

The location where PMUs are placed is critical: the network must be fully monitored the cost of monitoring must be minimized

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Problem

The location where PMUs are placed is critical: the network must be fully monitored the cost of monitoring must be minimized Definition (PMUs placing problem) Given an electric power network, find the locations where PMUs must be placed to monitor the entire network using the minimum number of PMUs possible.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Electrical Engineering to Graph Theory

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Problem

PMU Placing Problem → Power Domination Problem Engineering Graph Theory

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Problem

PMU Placing Problem → Power Domination Problem Engineering Graph Theory Power domination problem: Given a graph, find a set of vertices, with minimum cardinality, that can power dominate the entire graph after the iterated application of certain rules.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Problem

PMU Placing Problem → Power Domination Problem Engineering Graph Theory Power domination problem: Given a graph, find a set of vertices, with minimum cardinality, that can power dominate the entire graph after the iterated application of certain rules. The rules are defined so that placing a PMU at each vertex of a power dominating set suffices to monitor the power network.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Rule 1: Domination rule A vertex power dominates itself and its neighbors. Rule 2: Propagation rule If a power dominated vertex v has exactly one non-power dominated neighbor u, then v also power dominates u.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Rule 1: Domination rule A vertex power dominates itself and its neighbors. Rule 2: Propagation rule If a power dominated vertex v has exactly one non-power dominated neighbor u, then v also power dominates u. The propagation rule is applied iteratively until it does not produce new power dominated vertices.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Rule 1: Domination rule A vertex power dominates itself and its neighbors. Rule 2: Propagation rule If a power dominated vertex v has exactly one non-power dominated neighbor u, then v also power dominates u. The propagation rule is applied iteratively until it does not produce new power dominated vertices. Throughout this talk, power dominate = monitor = observe

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Notation

In this work all graphs are finite, undirected, connected and have no loops and/or multiple edges. Definition G = (V , E) graph, S ⊆ V , the family of sets Mi(S), i ∈ N, is defined recursively by: M0(S) := S M1(S) := N[S] = S ∪ N(S) While ∃v ∈ Mi(S) such that |N(v) \ Mi(S)| = 1, Mi+1(S) := Mi(S) ∪ N(v).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Notation

In this work all graphs are finite, undirected, connected and have no loops and/or multiple edges. Definition G = (V , E) graph, S ⊆ V , the family of sets Mi(S), i ∈ N, is defined recursively by: M0(S) := S M1(S) := N[S] = S ∪ N(S) While ∃v ∈ Mi(S) such that N(v) \ Mi(S) = {u}, Mi+1(S) := Mi(S) ∪ {u}.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

𝑤" 𝑤# 𝑤$ 𝑤% 𝑤& 𝑤' 𝑤( 𝑤)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

𝑤" 𝑤# 𝑤$ 𝑤% 𝑤& 𝑤' 𝑤( 𝑤) 𝑁+(𝑇) = 𝑇 = {𝑤&};

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

𝑤" 𝑤# 𝑤$ 𝑤% 𝑤& 𝑤' 𝑤( 𝑤) 𝑁+(𝑇) = 𝑇 = {𝑤&}; 𝑁"(𝑇) = 𝑇 = {𝑤&, 𝑤#, 𝑤', 𝑤&};

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

𝑤" 𝑤# 𝑤$ 𝑤% 𝑤& 𝑤' 𝑤( 𝑤) 𝑁+(𝑇) = 𝑇 = {𝑤&}; 𝑁"(𝑇) = 𝑇 = {𝑤&, 𝑤#, 𝑤', 𝑤&}; 𝑁#(𝑇) = {𝑤&, 𝑤#, 𝑤', 𝑤&, 𝑤", 𝑤(, 𝑤$};

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Power Domination Rules

𝑤" 𝑤# 𝑤$ 𝑤% 𝑤& 𝑤' 𝑤( 𝑤) 𝑁+(𝑇) = 𝑇 = {𝑤&}; 𝑁"(𝑇) = 𝑇 = {𝑤&, 𝑤#, 𝑤', 𝑤&}; 𝑁#(𝑇) = {𝑤&, 𝑤#, 𝑤', 𝑤&, 𝑤", 𝑤(, 𝑤$}; 𝑁&(𝑇) = {𝑤&, 𝑤#, 𝑤', 𝑤&, 𝑤", 𝑤(, 𝑤$, 𝑤)};

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 Mi(S) = V for some i ∈ N.

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 Mi(S) = V for some i ∈ N. Definition (Power domination number) G = (V , E) graph, γP(G) = min{|S| : S ⊆ V , ∃i ∈ N : Mi(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 Mi(S) = V for some i ∈ N. Definition (Power domination number) G = (V , E) graph, γP(G) = min{|S| : S ⊆ V , ∃i ∈ N : Mi(S) = V }. Definition (Power domination problem) G = (V , E) graph, find S ⊆ V such that γP(G) = S and Mi = 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| : M1(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| : M1(S) = V }. Theorem (Haynes et al. 2002) For every graph G, 1 ≤ γP(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| : M1(S) = V }. Theorem (Haynes et al. 2002) For every graph G, 1 ≤ γP(G) ≤ γ(G). Haynes et al. also characterized the extremal graphs G such that γP(G) = 1 and those graphs G such that γP(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 = (VG, EG) and H = (VH, EH) graphs, the Cartesian product GH has V (GH) = VG × VH and E(GH) = {(g, h)(g′, h′) : g = g′, hh′ ∈ EH or h = h′, gg′ ∈ EG} Definition (Rectangular grid) The rectangular n × m grid is PnPm where Pn and Pm are paths

• f 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: γP(PnPm) ≤ m+1

4

• if m ≡ 4 mod 8

m

4

• therwise

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: γP(PnPm) ≤ m+1

4

• if m ≡ 4 mod 8

m

4

• therwise

The following result requires more work: Theorem (Dorfling & Henning 2006) If n ≥ m ≥ 1, γP(PnPm) = m+1

4

• if m ≡ 4 mod 8

m

4

• therwise

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: γP(PnCm) ≤ min m + 1 4

• ,

n + 1 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: γP(PnCm) ≤ min m + 1 4

• ,

n + 1 2

• The following result requires more work:

Theorem (Barrera & DF 2011) If n ≥ 1 and m ≥ 3, γP(PnCm) = min m+1

4

• ,

n+1

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: γP(CnCm) ≤ n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

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: γP(CnCm) ≤ n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

The following result requires more work: Theorem (Barrera & DF 2011) If n ≥ m ≥ 3, γP(CnCm) = n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

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

• f 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

• f 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

• ne 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

• ne 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 γP(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 γP(G) = |S|: Z(G) ≤ γP(G)(∆ + 1) so Z(G) ∆ + 1

• ≤ γP(G)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Z(G) and γP(G)

Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆. Then, Z(G) ∆

• ≤ γP(G).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Z(G) and γP(G)

Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆. Then, Z(G) ∆

• ≤ γP(G).

Sketch of the proof: ∪s∈SN[s] − vs where vs is an arbitrary vertex 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 γP(G)

If {s, t} is a power dominating set, N[s] ∪ N[t] is a zero forcing set but (N[s] − {vs}) ∪ (N[t] − {vt}) is not.

s 𝑤" t 𝑤$

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Z(G) and γP(G)

However, (N[s] ∪ N[t]) − {vs} is still a zero forcing set. |(N[s] ∪ N[t]) − {vs}| = |N[s]| + |N[t] − {vt}| − 1 ≤ (∆ + 1) + ∆ − 1 and we still have a zero forcing set of cardinality at most 2∆.

s 𝑤" t 𝑤$

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Z(G) and γP(G)

Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆. Then,

• Z(G)

∆

• ≤ γP(G).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Z(G) and γP(G)

Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆. Then,

• Z(G)

∆

• ≤ γP(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 γP(G)

Theorem (Benson et al. 2015) Let G be a graph of maximum degree ∆. Then,

• Z(G)

∆

• ≤ γP(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 with nullity M = M(A). Then,

• M

∆

• ≤ γP(G).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Rectangular Grids

Let Mn×m be the n × m rectangular grid, 1 ≤ m ≤ n,

Z(Mn×m) 4

≤ γP(Mn×m)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Rectangular Grids

Let Mn×m be the n × m rectangular grid, 1 ≤ m ≤ n,

Z(Mn×m) 4

≤ γP(Mn×m) Z(Mn×m) = min{n, m}

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Rectangular Grids

Let Mn×m be the n × m rectangular grid, 1 ≤ m ≤ n,

Z(Mn×m) 4

≤ γP(Mn×m) Z(Mn×m) = min{n, m} m

4

• ≤ γP(Mn×m)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Rectangular Grids

Let Mn×m be the n × m rectangular grid, 1 ≤ m ≤ n,

Z(Mn×m) 4

≤ γP(Mn×m) Z(Mn×m) = min{n, m} m

4

• ≤ γP(Mn×m)

A simple construction shows γP(Mn×m) ≤ ⌈ m+1

4 ⌉ so,

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Rectangular Grids

Let Mn×m be the n × m rectangular grid, 1 ≤ m ≤ n,

Z(Mn×m) 4

≤ γP(Mn×m) Z(Mn×m) = min{n, m} m

4

• ≤ γP(Mn×m)

A simple construction shows γP(Mn×m) ≤ ⌈ m+1

4 ⌉ so,

γP(Mn×m) = m+1

4

• if m ≡ 4 mod 8

m

4

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

Let Cn,m be the cylinder obtained as PnCm, m ≥ 3, n ≥ 1, Z(Cn,m) = min{m, 2n}

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

Let Cn,m be the cylinder obtained as PnCm, m ≥ 3, n ≥ 1, Z(Cn,m) = min{m, 2n} ∆(Cn,m) = 4

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

Let Cn,m be the cylinder obtained as PnCm, m ≥ 3, n ≥ 1, Z(Cn,m) = min{m, 2n} ∆(Cn,m) = 4 γP(Cn,m) ≥ min{⌈ m

4 ⌉, ⌈ n 2⌉}

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

Let Cn,m be the cylinder obtained as PnCm, m ≥ 3, n ≥ 1, Z(Cn,m) = min{m, 2n} ∆(Cn,m) = 4 γP(Cn,m) ≥ min{⌈ m

4 ⌉, ⌈ n 2⌉}

The construction previously seen yields γP(Cn,m) ≤ min m + 1 4

• ,

n 2

• Daniela Ferrero

Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

Let Cn,m be the cylinder obtained as PnCm, m ≥ 3, n ≥ 1, Z(Cn,m) = min{m, 2n} ∆(Cn,m) = 4 γP(Cn,m) ≥ min{⌈ m

4 ⌉, ⌈ n 2⌉}

The construction previously seen yields γP(Cn,m) ≤ min m + 1 4

• ,

n 2

• Daniela Ferrero

Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

min m 4

• ,

n 2

• } ≤ γP(Cn,m) ≤ min

m + 1 4

• ,

n 2

• Daniela Ferrero

Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

min m 4

• ,

n 2

• } ≤ γP(Cn,m) ≤ min

m + 1 4

• ,

n 2

• Theorem (Benson et al. 2015)

γP(Cn,m) =    n

2

• if 2n ≤ m

m

4

• if 2n > m and m ≡ 0

mod 4 m

4

• r

m+1

4

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Cylinders

min m 4

• ,

n 2

• } ≤ γP(Cn,m) ≤ min

m + 1 4

• ,

n 2

• Theorem (Benson et al. 2015)

γP(Cn,m) =    n

2

• if 2n ≤ m

m

4

• if 2n > m and m ≡ 0

mod 4 m

4

• r

m+1

4

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Tori

Let Tn,m be the torus obtained as CnCm, m ≥ n ≥ 3. Theorem (Benson et al. 2015) Z(Tn,m) = 2n − 1 if m = n and n is odd 2n

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Tori

Let Tn,m be the torus obtained as CnCm, m ≥ n ≥ 3. Theorem (Benson et al. 2015) Z(Tn,m) = 2n − 1 if m = n and n is odd 2n

• therwise

Obviously, ∆(Tn,m) = 4 and

• 2n

4

• =
• n

2

• .

In addition, of n is odd

• 2n−1

4

• =
• n

2

• so we conclude

γP(Tn,m) ≥ n 2

• .

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Tori

From the power dominating sets for tori constructed as in the previous example, γP(Tn,m) ≤ n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

Theorem (Benson et al. 2015) γP(Tn,m) ≤ n

2

• r

n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Tori

From the power dominating sets for tori constructed as in the previous example, γP(Tn,m) ≤ n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

Theorem (Benson et al. 2015) γP(Tn,m) ≤ n

2

• r

n+1

2

• if n ≡ 2 mod 4

n

2

• therwise

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Definition (Lexicographic product) G = (VG, EG) and H = (VH, EH) graphs, the direct product G ∗ H has V (GH) = VG × VH and E(G ∗ H) = {(g, h)(g′, h′) : (g = g′ and hh′ ∈ EH) or gg′ ∈ EG} Example: ∗ =

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Definition (Lexicographic product) G = (VG, EG) and H = (VH, EH) graphs, the direct product G ∗ H has V (G ∗ H) = VG × VH and E(G ∗ H) = {(g, h)(g′, h′) : (g = g′ and hh′ ∈ EH) or gg′ ∈ EG} Example: ∗ =

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Z(G) ∆(G) ≤ γP(G) ⇐ ⇒ Z(G) ≤ ∆(G)γP(G)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Z(G) ∆(G) ≤ γP(G) ⇐ ⇒ Z(G) ≤ ∆(G)γP(G) γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise

(Chang et al. 2012)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Z(G) ∆(G) ≤ γP(G) ⇐ ⇒ Z(G) ≤ ∆(G)γP(G) γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise

(Chang et al. 2012)

• γ(G) is the minimum cardinality S ⊆ V s.t. N[S] = V
• γt(G) is the minimum cardinality of S ⊆ V s.t. N(S) = V

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Z(G) ∆(G) ≤ γP(G) ⇐ ⇒ Z(G) ≤ ∆(G)γP(G) γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise

(Chang et al. 2012)

• γ(G) is the minimum cardinality S ⊆ V s.t. N[S] = V
• γt(G) is the minimum cardinality of S ⊆ V s.t. N(S) = V

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise,

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise,

Z(G ∗ H) ≤ γP(G ∗ H)∆(G ∗ H).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise,

Z(G ∗ H) ≤ γP(G ∗ H)∆(G ∗ H). degG∗H(g, h) = (degG g)|V (H)| + degH h for any vertex (g, h) ∈ V (G ∗ H) ∆(G ∗ H) = ∆(G)|V (H)| + ∆(H).

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

γP(G ∗ H) = γ(G) if γP(H) = 1 γt(G)

• therwise,

Z(G ∗ H) ≤ γP(G ∗ H)∆(G ∗ H). degG∗H(g, h) = (degG g)|V (H)| + degH h for any vertex (g, h) ∈ V (G ∗ H) ∆(G ∗ H) = ∆(G)|V (H)| + ∆(H). Thus, Z(G ∗ H) ≤ γ(G)

• ∆(G)|V (H)| + ∆(H)
• if γP(H) = 1

γt(G)

• ∆(G)|V (H)| + ∆(H)
• therwise.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Assume G is dG-regular and H is dH-regular. If γP(H) = 1 and γ(G) = 1, then Z(G ∗ H) = dG|V (H)| + dH.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lexicographic product

Assume G is dG-regular and H is dH-regular. If γP(H) = 1 and γ(G) = 1, then Z(G ∗ H) = dG|V (H)| + dH. Example: If n ≥ 2 and m ≥ 3, Z(Kn ∗ Cm) = (n − 1)m + 2. (Chang et al. 2012)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Other lower bound

Theorem (Liao 2014) In a graph G of order n , diameter D and maximum degree ∆, γP(G) ≥

n D∆+1 and this bound is best possible.

The lower bound is attained by the 16 × m rectangular grid, for any integer m ≥ 2.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Other lower bound

Theorem (Liao 2014) In a graph G of order n , diameter D and maximum degree ∆, γP(G) ≥

n D∆+1 and this bound is best possible.

The lower bound is attained by the 16 × m rectangular grid, for any integer m ≥ 2. Beautiful expression for a lower bound

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Other lower bound

Theorem (Liao 2014) In a graph G of order n , diameter D and maximum degree ∆, γP(G) ≥

n D∆+1 and this bound is best possible.

The lower bound is attained by the 16 × m rectangular grid, for any integer m ≥ 2. Beautiful expression for a lower bound...but it is wrong!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Counterexample

Theorem (Liao 2014) In a graph G of order n , diameter D and maximum degree ∆, γP(G) ≥

n D∆+1 and this bound is best possible.

In the example below, the bound is 82

37 while γP = 2

2 3 4 5 6 7 8 9 10 1 11 82

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Counterexample

Given ∆ ≥ 3, construct H∆ with three levels of vertices as follows:

• Level 1: One single vertex.
• Level 2: ∆ vertices adjacent with the vertex in level 1.
• Level 3: For each level 2 vertex, add ∆ − 1 vertices adjacent to it.
• Add edges to make a path along level 3.

γP(H∆) = 2 while the bound is asymptotically ∆

4 .

2 3 4 5 6 7 8 9 10 1 11 82

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. In each transition Mi(S) − → Mi+1(S) at most ∆|S| can be added.

𝑁"(𝑇) =𝑇 𝑁&(𝑇) =𝑂[𝑇] 𝑁*(𝑇) 𝑁+(𝑇) = 𝑊(𝐻) |𝑊| ≤ |𝑇| + ∆|𝑇|𝑚

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). Then, |V | ≤ |S| + ℓ∆|S| = |S|(1 + ℓ∆) If S is a minimal power dominating set γP(G) = |S| then |V | ≤ γP(G)(1 + ℓ∆) so |V | 1 + ℓ∆ ≤ γP(G)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know:

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know: For every vertex v in V there exists a trail joining a vertex in S with v.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know: For every vertex v in V there exists a trail joining a vertex in S with v. The length of a trail is not upper bounded by the diameter.

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Correction

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know: For every vertex v in V there exists a trail joining a vertex in S with v. The length of a trail is not upper bounded by the diameter. Except when the graph is a tree....

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lower bound for trees

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know:

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lower bound for trees

Let us look at the propagation process for a graph G and a given power dominating set S. Let ℓ be the minimum positive integer i such that Mi(S) = V (G). We know: Theorem (DF et al. 2015) Let T be a tree with order n, maximum degree ∆ and diameter D. Then, n 1 + (D − 1)∆ ≤ γP(T)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Lower bound for trees

Theorem (DF et al. 2015) Let T be a tree with order n, maximum degree ∆ and diameter D. Then, n 1 + (D − 1)∆ ≤ γP(T) Observe that Liao’s bound for trees was n 1 + D∆ ≤ γP(T) while we obtained n 1 + (D − 1)∆ ≤ γP(T)

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Related problems

Forbidden zone

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Related problems

Forbidden zone Fault-tolerance

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Related problems

Forbidden zone Fault-tolerance Parametrized complexity

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

Related problems

Forbidden zone Fault-tolerance Parametrized complexity Propagation number

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem

Introduction Exact results Bounds Conclusion

The end

Thank you!

Daniela Ferrero Lower Bounds for the Power Domination Problem