Note on Nordhaus-Gaddum problems for power domination

Transcription

1 Note o Nordhaus-Gaddum problems for power domiatio Katherie F. Beso Departmet of Mathematics ad Physics Westmister College Fulto, MO 65251, USA katie.beso@westmister-mo.edu Mary Flagg Departmet of Mathematics, Computer Sciece ad Cooperative Egieerig Uiversity of St. Thomas 800 Motrose, Housto, TX 77006, USA flaggm@stthom.edu Leslie Hogbe Departmet of Mathematics Iowa State Uiversity Ames, IA 50011, USA America Istitute of Mathematics 600 E. Brokaw Road, Sa Jose, CA 95112, USA hogbe@aimath.org Daiela Ferrero Departmet of Mathematics Texas State Uiversity Sa Marcos, TX 78666, USA dferrero@txstate.edu Veroika Furst Departmet of Mathematics Fort Lewis College Durago, CO 8101, USA furst v@fortlewis.edu Violeta Vasilevska Departmet of Mathematics Utah Valley Uiversity Orem, UT, 8058, USA Violeta.Vasilevska@uvu.edu Submitted: October 10, 2016; Accepted: April 28, 2017; Published: XX Mathematics Subject Classificatios: 05C69, 05C57 Abstract The upper ad lower Nordhaus-Gaddum bouds over all graphs for the power domiatio umber follow from kow bouds o the domiatio umber ad examples. I this ote we improve the upper sum boud for the power domiatio umber substatially for graphs havig the property that both the graph ad its complemet must be coected. For these graphs, our boud is tight ad is also sigificatly better tha the correspodig boud for domiatio umber. We also improve the product upper boud for the power domiatio umber for graphs with certai properties. Keywords: power domiatio; domiatio; zero forcig; Nordhaus-Gaddum Supported by NSF DMS the electroic joural of combiatorics 22 (2015), #P00 1

2 1 Itroductio The study of the power domiatio umber of a graph arose from the questio of how to moitor electric power etworks at miimum cost, see Hayes et al. [9]. Ituitively, the power domiatio problem cosists of fidig a set of vertices i a graph that ca observe the etire graph accordig to certai observatio rules. The formal defiitio is give below immediately after some graph theory termiology. A graph G = (V, E) is a ordered pair formed by a fiite oempty set of vertices V = V (G) ad a set of edges E = E(G) cotaiig uordered pairs of distict vertices (that is, all graphs are simple ad udirected). The complemet of G = (V, E) is the graph G = (V, E), where E cosists of all two elemet subsets of V that are ot i E. For ay vertex v V, the eighborhood of v is the set N(v) = {u V : {u, v} E} ad the closed eighborhood of v is the set N[v] = N(v) {v}. Similarly, for ay set of vertices S, N(S) = v S N(v) ad N[S] = v S N[v]. For a set S of vertices i a graph G, defie P D(S) V (G) recursively as follows: 1. P D(S) := N[S] = S N(S). 2. While there exists v P D(S) such that N(v) \ P D(S) = 1: P D(S) := P D(S) N(v). A set S V (G) is called a power domiatig set of a graph G if, at the ed of the process above, P D(S) = V (G). A miimum power domiatig set is a power domiatig set of miimum cardiality. The power domiatio umber of G, deoted by γ P (G), is the cardiality of a miimum power domiatig set. Power domiatio is aturally related to domiatio ad to zero forcig. A set S V (G) is called a domiatig set of a graph G if N[S] = V (G). A miimum domiatig set is a domiatig set of miimum cardiality. The domiatio umber of G, deoted by γ(g), is the cardiality of a miimum domiatig set. Clearly γ P (G) γ(g). Zero forcig was itroduced idepedetly i combiatorial matrix theory [1] ad cotrol of quatum systems [5]. From a graph theory poit of view, zero forcig is a colorig game o a graph played accordig to the color chage rule: If u is a blue vertex ad exactly oe eighbor w of u is white, the chage the color of w to blue. We say u forces w. A zero forcig set for G is a subset of vertices B such that whe the vertices i B are colored blue ad the remaiig vertices are colored white iitially, repeated applicatio of the color chage rule ca color all vertices of G blue. A miimum zero forcig set is a zero forcig set of miimum cardiality. The zero forcig umber of G, deoted by Z(G), is the cardiality of a miimum zero forcig set. Power domiatio ca be see as a domiatio step followed by a zero forcig process, ad we will use the termiology v forces w to refer to Step 2 of power domiatio. Clearly γ P (G) Z(G). For a graph parameter ζ, the followig are Nordhaus-Gaddum problems: Determie a (tight) lower or upper boud o ζ(g) + ζ(g). Determie a (tight) lower or upper boud o ζ(g) ζ(g). the electroic joural of combiatorics 22 (2015), #P00 2

3 The ame comes from the ext theorem of Nordhaus ad Gaddum, where χ(g) deotes the chromatic umber of G. Theorem 1. [16] For ay graph G of order, ad 2 χ(g) + χ(g) + 1 ( ) χ(g) χ(g). 2 Each boud is assumed for ifiitely may values of. Nordhaus-Gaddum bouds have bee foud for both domiatio ad zero forcig. I additio to the origial papers cited here, Nordhaus-Gaddum results for domiatio ad several variats (but ot power domiatio) are discussed i Sectio 9.1 of the book [10] ad i the survey paper [2]. Theorem 2. [1] For ay graph G of order 2, γ(g) + γ(g) + 1 ad 2 γ(g) γ(g). The upper bouds are realized by the complete graph K, ad the lower bouds are realized by the star (complete bipartite graph) K 1, 1. ad It is kow that for a graph G of order 2, 2 Z(G) + Z(G) 2 1 Z(G) Z(G) 2, with the upper bouds realized by the complete graph K ad the lower bouds realized by the path P for. That the upper bouds are correct is immediate. The result 2 Z(G) + Z(G) appears i [7]. The Z(G) Z(G) follows, because 1 Z(G) for all G ad the fuctio f(z) = z( 2 z) attais its miimum o the iterval [1, ] at the edpoits. The geeral Nordhaus-Gaddum upper bouds for power domiatio umber follow from those for domiatio umber give i Theorem 2. The iequalities 2 γ P (G) + γ P (G) ad 1 γ P (G) γ P (G) are obvious sice 1 γ P (G) for every graph, ad these are realized by the path P (it is straightforward to verify that γ P (P ) = 1 = γ P (P )). Corollary. For ay graph G of order, 2 γ P (G) + γ P (G) + 1 ad 1 γ P (G) γ P (G). The upper bouds are realized by the complete graph K, ad the lower bouds are realized by the path P. the electroic joural of combiatorics 22 (2015), #P00

4 I Sectio we improve the sum upper boud for the power domiatio umber sigificatly uder the assumptio that both G ad G are coected, or more geerally all compoets of both have order at least, ad show that this boud is substatially differet from the aalogous boud for domiatio umber. I Sectio we refie the product bouds for certai special cases. Sectio 2 cotais additioal results that we use i Sectios ad. Sectio 5 summarizes the bouds for domiatio umber, power domiatio umber, ad zero forcig umber. Some additioal otatio is used: Let K p,q deote a complete bipartite graph with partite sets of cardiality p ad q. The degree of vertex v is deg G v = N G (v). Let δ(g) (respectively, (G)) deote the miimum (respectively, maximum) of the degrees of the vertices of G. A cut-set is a set of vertices whose removal discoects G. The vertexcoectivity of G K, deoted by κ(g), is the miimum cardiality of a cut-set (ote κ(g) = 0 if G is discoected), ad κ(k ) = 1. A edge-cut is a set of edges whose removal discoects G, ad the edge-coectivity of G, deoted by λ(g), is the miimum cardiality of a edge-cut. Observe that κ(g) λ(g) δ(g). The distace betwee vertices u ad v i G, d G (u, v), is the legth of a shortest path betwee u ad v i G. The diameter of G, diam(g), is the maximum distace betwee two vertices i a coected graph G; diam(g) = if G is ot coected. A compoet of a graph is a maximal coected subgraph. 2 Tools for Nordhaus-Gaddum bouds for power domiatio I this sectio we establish results that will be applied to improve Nordhaus-Gaddum upper bouds for both the sum ad product of the power domiatio umber with additioal assumptios, such as every compoet of the graph ad its complemet has order at least. The ext result is immediate from Corollary. Corollary. For ay graph G of order, γ P (G) γ P (G) Next we cosider the relatioship betwee the power domiatio umber of G or G ad the miimum degree or vertex-coectivity of G. Remark 5. For ay graph G of order, γ(g) δ(g) + 1, because a vertex of maximum degree i G, which is 1 δ(g), together with all its o-eighbors is a domiatig set of G. Propositio 6. Let G be a graph such that either G or G has isolated vertices. The γ P (G) δ(g). If δ(g) = 1, the γ P (G) = 1. Proof. Costruct a power domiatig set S for G of cardiality δ(g) as follows: Put a vertex v of maximum degree i G ito S, so N G [v] = (G) + 1 = 1 δ(g) + 1 = δ(g) <, where is the order of G. The add all but oe of the vertices i V (G) \ N G [v] ito S, i.e., add δ(g) 1 0 vertices to S, so S = δ(g). Now N G [S] cotais all but at most oe vertex, ad sice G has o isolated vertices, ay eighbor of such a vertex ca force it. The last statemet the follows sice γ P (G) 1 for all graphs G.. the electroic joural of combiatorics 22 (2015), #P00

5 Theorem 7. [11] If G is a graph with diam(g) = 2, the γ(g) κ(g). Next we state several results that give sufficiet coditios for γ(g) 2 or γ(g) 2, which the imply γ P (G) 2 or γ P (G) 2. Theorem 8. [], [10, Theorem 2.25] If G is a graph with diam(g), the γ(g) 2. Note that Theorem 8 also applies to graphs that are ot coected. Theorem 9. Suppose G is a graph with diam(g) = 2 such that G has o isolated vertices. The γ P (G) κ(g) 1 or γ P (G) 2. Proof. Sice G has o isolated vertices, every vertex has a eighbor i G. Let S be a miimum cut-set for G. Sice diam(g) = 2, every vertex i V \ S is adjacet to at least oe vertex i S. Case 1: There exists a vertex u V \ S that is adjacet to exactly oe vertex i S, say v (Case 1 is the oly possible case whe κ(g) = 1). Let G 1 deote the compoet of G S cotaiig u. I G, u domiates S \ {v} ad all vertices i compoets of G S other tha G 1. Let x be ay vertex i a compoet of G S that is ot equal to G 1. The x domiates the vertices of G 1. Therefore, {u, x} domiates all vertices i V except possibly v, ad ay eighbor of v i G ca force v, so {u, x} is a power domiatig set for G. Thus, γ P (G) 2. Case 2: Every vertex i V \ S is adjacet to at least two vertices i S. The S \ {v} is a power domiatig set for ay vertex v S, because S \ {v} domiates V \ {v}, ad ay eighbor of v i G ca force v. Thus, γ P (G) κ(g) 1. Theorem 10. [8] If G is plaar ad diam(g) = 2, the γ(g) 2 or G = S (K ), the graph show i Figure 1. Furthermore, γ(s (K )) =. Figure 1: The graph S (K ), which is the oly plaar graph with diameter 2 ad domiatio umber greater tha 2. Corollary 11. If G is plaar ad diam(g) = 2, the γ P (G) 2. Proof. This follows from Theorem 10 ad the fact that γ P (S (K )) = 2. the electroic joural of combiatorics 22 (2015), #P00 5

6 Whe κ(g) = λ(g) = δ(g), G is maximally coected. I every maximally coected graph G, for ay vertex v such that deg v = δ(g), N G (v) is a miimum cut-set ad the set of all edges icidet with v is a miimum edge-cut. I this case we say the cut is trivial, because it leaves a coected compoet formed by oe isolated vertex. A maximally coected graph G is super-λ if every miimum edge-cut is trivial. Super-λ graphs of diameter 2 were characterized by Wag ad Li: Theorem 12. [19] A coected graph G with diam(g) = 2 is super-λ if ad oly if G cotais o subgraph K δ(g) i which all of the vertices have degree (i G) equal to δ(g). Propositio 1. Let G be a coected graph with diam(g) = 2. If G is ot super-λ, the γ P (G) 2. Proof. Sice G is ot super-λ, there exists a subgraph K δ(g) i which all of the vertices have degree equal to δ(g) i G. Let v be a vertex i this K δ(g), so v has exactly oe eighbor outside K δ(g), say w. The, v domiates all vertices i K δ(g) ad w. Sice every vertex u i K δ(g) has degree i G equal to δ(g) ad u has δ(g) 1 domiated eighbors, u ca force its oe remaiig eighbor. Therefore, all vertices i K δ(g) ad their eighbors are observed. Sice diam(g) = 2, d(v, x) = 1 or d(v, x) = 2 for every vertex x v i G. If d(v, x) = 1, the x is domiated by v. If d(v, x) = 2, the x is a eighbor of a vertex i N G (v). Sice the vertices i N G (v) that are i K δ(g) have forced their eighbors, the oly case i which x is ot observed is if it is a eighbor of w. Thus {v, w} is a power domiatig set. Corollary 1. Assume that G ad G both have all compoets of order at least. The γ P (G) 2 or γ P (G) 2 if ay of the coditios below is satisfied: 1. diam(g) or diam(g). 2. G or G is plaar.. κ(g) or κ(g).. G or G is ot super-λ. Proof. Part (1) follows from Theorem 8. Sice G ad G both have all compoets of order at least, diam(g) 1 ad diam(g) 1. The case diam(g) is covered by part (1). So assume diam(g) = 2. The (2), (), ad () follow from Corollary 11, Theorem 9, ad Propositio 1, respectively. Let T be the family of graphs costructed by startig with a coected graph H ad for each v V (H) addig two ew vertices v ad v, each adjacet to v ad possibly to each other but ot to ay other vertices. The ext result appears i [21] without the floor fuctio. Theorem 15. [21] Suppose every compoet of a graph G has order at least ad deotes the order of G. The γ P (G). Furthermore, if γp (G) =, the every compoet of G is i T {K, }. the electroic joural of combiatorics 22 (2015), #P00 6

7 The method used i the costructio of a graph G T implies that γ P (G) = 1 if we start with a graph o at least 2 vertices: Lemma 16. Suppose G is a graph havig distict vertices w, u, v, v, ad v such that N[v ] = N[v ] = {v, v, v }, u N(v) ad w N(v). The γ P (G) = 1. Proof. I G, u is ot adjacet to v but is adjacet to v ad to v. The {v } is a power domiatig set for G, because u N G [v ] = V (G) \ {v, v } ad u forces v i G, ad the w forces v i G. Thus γ p (G) = 1. Propositio 17. Suppose G is a graph of order such that every compoet of G ad G has order at least ad γ P (G) =. The γ P (G) 2. If, i additio, G has a compoet G o T of order at least 6, the γ P (G) = 1. Proof. Necessarily, is a multiple of ad. If G has 2 or more compoets, the γ(g) 2 by Theorem 8. If G = K,, the γ P (G) = 2. Now suppose G has a compoet G o T of order at least 6 (this icludes the case where G has oly oe compoet that is ot K, ). The γ P (G o ) = 1 by Lemma 16 ad Propositio 6 (for the case v N(v )). I G, ay vertex i G o domiates ay vertex i a differet compoet, so the oe vertex that power domiates G o also power domiates G, ad γ P (G) = 1. Theorem 18. [11, 1] Suppose G is a graph of order with diam(g) = 2. If 2, the γ(g), ad γ(g) + 1 for 2. Remark 19. Let G be a graph. Suppose W is a set of at least two vertices such that o vertex outside W is adjacet to exactly oe vertex i W. The every power domiatig set S must cotai a vertex i N[W ], because o vertex outside of W ca force a vertex i W uless all but oe of the vertices i W have already bee power domiated. Figure 2: The ecklace N The ecklace with s diamods, deoted N s, is a -regular graph that ca be costructed from a s-cycle by appedig s additioal vertices, with each ew vertex adjacet to sequetial cycle vertices; N is show i Figure 2. Theorem 20. [6] Suppose G is a coected -regular graph of order ad G K,. The γ P (G), ad this boud is attaied for arbitrarily large by G = Nr. the electroic joural of combiatorics 22 (2015), #P00 7

8 Lemma 21. For r 2, γ P (N r ) = 2. Proof. Ay two vertices that are i differet copies of K e ad are ot icidet to the missig edges domiate N r, so γ P (N r ) 2. To complete the proof, we show that o oe vertex v ca power domiate N r. Deote the vertices of the K e that cotais v by x, y, z, w, where e = {x, y}. Apply Remark 19 to W = {z, w} for v = x ad to W = {x, y, w} for v = z to coclude {v} is ot a power domiatig set; the cases v = y or w are similar. Nordhaus-Gaddum sum bouds for power domiatio I this sectio, we improve the tight Nordhaus-Gaddum sum upper boud of for all graphs (Corollary ) to approximately uder oe of the assumptios that each compoet of G ad G has order at least (Theorem 2 below), or that both G ad G are coected (Theorem 25 below), ad to approximately i some special cases. The lower boud 2 γ P (G) + γ P (G) ca be attaied with both G ad G coected, specifically by the path G = P (both P ad P are coected for ). But the upper boud for all graphs is attaiable oly by discoectig G or G with some very small compoets. The ext result follows from Corollary 1 ad Theorem 15. Corollary 22. Let G be a graph of order such that every compoet of G ad G has order at least ad (diam(g) or diam(g) or κ(g) or κ(g) ). The γ P (G) + γ P (G) + 2. Theorem 2. Suppose G is a graph of order such that every compoet of G ad G has order at least. The for 1, 1, 16, 17, 20, γ P (G) + γ P (G) + 2, ad this boud is attaied for arbitrarily large by G = rk (where r 2). For = 1, 1, 16, 17, 20, γ P (G) + γ P (G) +. Proof. Without loss of geerality, we assume γ P (G) γ P (G), ad let p = γ P (G) ad p = γ P (G). If p 2, the p + p + 2 follows from Theorem 15. If p 6, Corollary gives p + p p +. So we assume p 5. Sice diam(g), diam(g) 1, by p Corollary 22 we may also assume diam(g) = diam(g) = 2 ad κ(g), κ(g). The latter implies 9. Corollary implies p + p p +. By Theorem 18, p, p p + 1. Thus we eed to cosider the followig cases: p =,, i which case p + p +. p = 5, i which case p + p the electroic joural of combiatorics 22 (2015), #P00 8

9 Algebra shows that ad for 21 ad = 18, 19. For = 9, 10, 11, p + p 5 = + 2 has bee verified computatioally [12]. To complete the proof that p + p + 2 for 1, 1, 16, 17, 20, we cosider = 12, 15. Sice p p, the oly possibilities are = 12 with (p, p) = (, ), (, ), p or = 15 with (p, p) = (, ), (, ), (, 5). For = 12 with (p, p) = (, ), ad = 15 with (p, p) = (, ), (, ), γ P (G) + γ P (G) + 2. I each of the remaiig cases, = 12 with (p, p) = (, ), or = 15 with (p, p) = (, 5), observe that p = ad p =. But this is prohibited by Propositio 17. If G is a disjoit uio of r 2 copies of K, the γ P (G) + γ P (G) = + 2, so the boud is tight for arbitrarily large. Fially, cosider = 1, 1, 16, 17, 20. For p =, γ P (G) + γ P (G) + is immediate from Theorem 15. Sice p p, the oly remaiig cases are = 16 p or 17 with (p, p) = (, ), ad = 20 with (p, p) = (, ), (, 5). All of these satisfy γ P (G) + γ P (G) +. We have o examples cotradictig γ P (G) + γ P (G) + 2 for graphs G of ay order where the order of each compoet of G ad G is at least. We cojecture that these exceptioal values 1, 1, 16, 17, 20 of are ot i fact exceptios: Cojecture 2. If G is graph of order such that the order of each compoet of G ad G is at least, the γ P (G) + γ P (G) + 2. Next we cosider the case i which both G ad G are required to be coected. Theorem 25. Suppose G is a graph of order such that both G ad G are coected. The for 12, 1, 1, 15, 16, 17, 18, 20, 21, 2, γ P (G) + γ P (G) + 1, ad this boud is attaied for arbitrarily large 6 by G T. Proof. For ot a multiple of, + 1 = + 2, ad the result follows from Theorem 2. So assume is a multiple of. We proceed as i the proof of Theorem 2, with the same otatioal covetios p := γ P (G) p := γ P (G), ad agai the boud is established for p 6 by Corollary. The result follows from Theorem 15 for p = 1, so we assume 2 p 5. Because is a multiple of ad both G ad G are coected, 6. Sice K, is ot coected, Theorem 15 ad Propositio 17 prohibit p 2 ad p =, so p 1. Thus the result follows for p = 2. So we assume p 5 ad p p 1; the latter requires 12, ad sice all multiples of betwee 12 ad 26 are excluded by hypothesis, we may assume 27. Sice p, diam(g) 2 by Theorem 8. Sice diam(g) = 1 would imply p = 1, ecessarily diam(g) = 2. The p by Theorem 18. Hece p + p + for p =,, ad p + p for p = 5. Algebra shows that ad for 27 ad a multiple of. There are graphs G T of arbitrarily large order, G is coected for 6, ad these graphs attai the boud. the electroic joural of combiatorics 22 (2015), #P00 9

10 The tight upper boud i Theorem 25 for γ P (G)+γ P (G) with both G ad G coected was obtaied by switchig from floor to ceilig. This raises a questio about the boud with floor, which has implicatios for products (see Sectio ). Questio 26. Do there exist graphs G of arbitrarily large order with both G ad G coected such that γ P (G) + γ P (G) = + 2? The ext two examples, foud via the computer program Sage, show that there are pairs of coected graphs G ad G of orders = 8 ad 11 such that γ P (G) + γ P (G) = + 2. Example 27. Let G be the graph show with its complemet i Figure ; observe that both are coected. It is easy to see that o oe vertex power domiates either G or G ad also easy to fid a power domiatig set of two vertices for each. Thus 8 γ P (G) + γ P (G) = = = + 2. Figure : A coected graph G of order 8 ad its coected complemet G such that γ P (G) + γ P (G) = + 2. Example 28. Let G be the graph show i Figure. It is easy to see that G is also coected. First we show that o set of two vertices is a power domiatig set for G. Sice {1, 2, 7} is a power domiatig set for G, this will imply γ P (G) = = 11. By Remark 19 applied to the sets W 1 = {2, } ad W 2 = {7, 8}, ay power domiatig set S of G must cotai vertices u 1 {2,,, 5, 6} ad aalogously, u 2 {7, 8, 9, 10, 11}. If u 1 {2, } ad u 2 {7, 8}, the vertex 1 caot be forced. If u 1 {, 5, 6}, the the two remaiig vertices i {, 5, 6} caot be forced; the case i which u 2 {9, 10, 11} is symmetric. Next we show that o oe vertex is a power domiatig set for G. Sice {2, 7} is a power domiatig set for G, this will imply γ P (G) = 2 ad γ P (G) + γ P (G) = For each possible vertex v {1, 2,,, 5, 6}, we apply Remark 19 with W as show: For v {1, 2, }, use W = {, 5, 6}. For v {, 5, 6}, use W = {2, }. The case v {7, 8, 9, 10, 11} is symmetric. The ext two theorems for domiatio umber provide a iterestig compariso. Theorem 29. [] For ay graph G of order such that δ(g) 1 ad δ(g) 1, γ(g) + γ(g) + 2, 2 ad this boud is attaied for arbitrarily large. the electroic joural of combiatorics 22 (2015), #P00 10

11 Figure : A coected graph G of order 11 such that G is also coected ad γ P (G) + γ P (G) = + 2. Theorem 0. [11] Suppose G is a graph of order such that δ(g) 7 ad δ(g) 7. The γ(g) + γ(g) + 2. From Theorem 0 we see that the same sum upper boud we obtaied for power domiatio umber (with the weaker hypothesis that every compoet has order at least ) is obtaied for domiatio umber whe we make the stroger assumptio that the miimum degrees of both G ad G are at least 7. Theorem 29 is a more direct parallel to Theorem 2 but with a higher boud. Theorem 29 has a weaker hypothesis, which is equivalet to every compoet of G ad G has order at least 2. The ext example shows that if Theorem 29 is restated to require both G ad G to be coected, the boud remais tight. This provides a direct compariso with Theorem 25 ad shows that for graphs G with both G ad G coected, the upper boud for the domiatio sum is substatially higher tha the upper boud for the power domiatio sum. Example 1. Let G k deote the kth comb, costructed by addig a leaf to every vertex of a path P k (G 9 is show i Figure 5); the order of G k is 2k. The every domiatig set S must have at least k elemets, because for each of the k leaves, either the leaf or its eighbor must be i S. Sice two vertices are eeded to domiate G k, γ(g k ) + γ(g k ) = k + 2 = 2k + 2. The results for power domiatio are very differet. For k = s, oe third 2 of the vertices i P k ca power domiate G k, ad oe vertex ca power domiate G k, so γ P (G k ) + γ P (G k ) = s + 1 = 2k Figure 5: The comb G 9 with the vertices of a miimum power domiatig set colored. We ca also improve the boud i Corollary whe G has some compoets of order less tha ad G has at least oe edge. the electroic joural of combiatorics 22 (2015), #P00 11

12 Theorem 2. Let G be a graph of order that has 1 isolated vertices ad 2 copies of K 2 as compoets such that 1 ad 2 are ot both zero. The γ P (G) + γ P (G) Proof. As a cosequece of Theorem 15, ( ) γ P (G) Because 1 1 or 2 1, a isolated vertex (respectively, oe of the vertices i a K 2 compoet) power domiates the complemet, so γ P (G) = 1. Hece, ( ) γ P (G) + γ P (G) We ca also improve the upper boud i some special cases. Theorem. Suppose G is a graph of order with diam(g) = diam(g) = 2, ad oe of the followig is true: 1. G or G is plaar. 2. κ(g) or κ(g).. G or G is ot super-λ. If 2, the γ P (G) + γ P (G) + 2, ad γp (G) + γ P (G) + for 2. Proof. By Corollary 1, γ P (G) 2 or γ P (G) 2. Assume without loss of geerality that γ P (G) 2. Applyig Theorem 18 to G, γ P (G) for 2 ad γp (G) + 1 for 2. Theorem. Suppose G is a -regular graph of order 6 such that o compoet is K,. The γ P (G), γp (G) 2, ad γ P (G) + γ P (G) + 2, ad all these iequalities are tight for arbitrarily large. Proof. Suppose first that G is coected. The γ P (G) by Theorem 20 (sice G K, ), so it suffices to show γ P (G) 2. Sice G K ad G is -regular, diam(g) 2. Sice diam(g) implies γ P (G) 2 by Theorem 8, we assume diam(g) = 2. For ay vertex v, there are at most 10 vertices at distace 0, 1, or 2 from v (v, its eighbors, ad two additioal eighbors of each of the eighbors of v), so 10. A examiatio of -regular graphs with 6 10 (see, for example, [18, p. 127]) shows the oly such graphs of diameter 2 are the five graphs show i Figure 6 (amed as i [18]): C = K,, C2, C5, C7, ad C27 (the Peterse graph). It is straightforward to verify that γ P (G) = 1 for G {C2, C7} ad γ P (G) = 2 for G {C5, C27}. This completes the proof for the case i which G is coected. the electroic joural of combiatorics 22 (2015), #P00 12

13 Figure 6: The five cubic graphs of diameter 2: C = K,, C2, C5, C7, ad C27 = the Peterse graph. Now assume G has compoets G 1,..., G s with s 2. The γ P (G) 2 by Theorem 8. Sice G i K,, γ P (G) = s γ P (G i ) i=1 s i=1 i s i=1 i = The graphs N r attai the boud by Theorem 20 ad Lemma 21.. Nordhaus-Gaddum product bouds for power domiatio As with the sum, the tight product lower boud for the power domiatio umber for all graphs G remais uchaged eve with the additioal requiremet that both G ad G be coected (usig the path). I Sectio, we achieved a tight sum upper boud for such graphs. However, sice this was achieved with γ P (G) = 1 for both G ad G coected, ad with γ P (G) = 2 whe each compoet of both G ad G has order at least, there are few immediate implicatios for products (see Sectio 5 for further discussio of coectios betwee sum ad product bouds). Questio 5. Does there exist a graph G of order such that all compoets of G ad G have order at least ad γ P (G) γ P (G) > 2? Remark 6. If the aswer to Questio 5 is egative, the the graphs G = rk with r 2 show 2 is a tight upper boud for the product, because γp (G) = ad γ P (G) = 2. Remark 7. If the aswer to Questio 26 is positive, the such graphs show 2 ca be attaied for arbitrarily large for the product with both G ad G coected. We ca improve the product boud i certai special cases. The ext result follows from Corollary 1 ad Theorem 15. Corollary 8. Let G be a graph of order such that every compoet of G ad G has order at least. The γ P (G) γ P (G) 2 if at least oe of the followig is true: 1. diam(g) or diam(g). 2. G or G is plaar.. κ(g) or κ(g). the electroic joural of combiatorics 22 (2015), #P00 1

14 . G or G is ot super-λ. The ext two results are product aalogs of Theorems 2 ad. The proofs, which are aalogous, are omitted. Theorem 9. Let G be a graph of order that has 1 isolated vertices ad 2 copies of K 2 as compoets such that 1 ad 2 are ot both zero. The γ P (G) γ P (G) Theorem 0. Suppose G is a graph of order with diam(g) = diam(g) = 2, ad oe of the followig is true: 1. G or G is plaar. 2. κ(g) or κ(g).. G or G is ot super-λ. If 2, the γ P (G) γ P (G) 2, ad γp (G) γ P (G) for 2. The ext result follows immediately from Theorem. Corollary 1. Suppose G is a -regular graph of order 6 with o K, compoet. The γ P (G) γ P (G) 2, ad this boud is attaied for arbitrarily large. Propositio 2. Let G be a tree o vertices. If G is ot K 1, or K 1,, the γ P (G) γ P (G) ad this boud is attaied for arbitrarily large. Proof. Note first that sice G is coected, γ P (G) by Theorem 15. If a tree is ot a star, the its complemet is also coected, ad by Propositio 6, γ P (G) = 1. For a star graph K 1, 1, we have γ P (K 1, 1 ) γ P (K 1, 1 ) = 2, which is less tha or equal to whe 6. The boud is attaied for arbitrarily large because if G is costructed from ay tree T by addig two leaves to each vertex of T, the γ P (G) =. 5 Summary ad discussio Table 1 summarizes what is kow about Nordhaus-Gaddum sum bouds for power domiatio umber, domiatio umber, ad zero forcig umber. Both the sum ad product upper ad lower bouds for the domiatio umber were determied by Jaeger ad Paya i 1972 (see Theorem 2), ad aalogous bouds for power domiatio are immediate corollaries. Sice the, there have bee umerous improvemets to the sum upper boud for domiatio umber uder various coditios o the electroic joural of combiatorics 22 (2015), #P00 1

15 Table 1: Summary of tight bouds for ζ(g) + ζ(g) for ζ = γ P, γ, Z ζ & restrictios lower upper γ P γ P & all compoets of both G ad G of order & γ P & both G ad G coected & γ & γ & both G ad G coected + 2 Z & Z & both coected 2 2 o() 2 G ad G. Examples of such coditios iclude requirig every compoet of both G ad G to have order at least 2 or requirig both to be coected or requirig both to have miimum degree at least 7. I Sectio we established better upper bouds for the power domiatio umber i the cases where both G ad G are coected or both have every compoet of order at least. By cotrast, results o products are very sparse for both domiatio umber ad power domiatio umber. Historically, the Nordhaus-Gaddum sum upper boud has ofte bee determied first, ad the used to obtai the product upper boud, as i the case of Nordhaus ad Gaddum s origial results [16] (see Theorem 1). I order to use this techique of gettig a tight product boud from a tight sum boud, oe eeds the sum upper boud to be optimized with approximately equal values or the sum lower boud to be optimized o extreme values. The sum lower boud for the domiatio umber is optimized at the extreme values, ad therefore the tight lower boud for the sum yields a tight lower boud for the product. However, all available evidece suggests that, for both the domiatio umber ad the power domiatio umber, the sum upper boud is optimized oly at extreme values. For example, the sum upper boud of + 1 over all graphs is attaied oly by the values 1 ad for both the domiatio ad power domiatio umbers. Thus, for the domiatio umber ad the power domiatio umber, the Nordhaus-Gaddum product upper boud presets challeges. Further evidece idicatig that the sum boud is optimized oly o extreme values comes from radom graphs. Ad it is also iterestig to cosider the average behavior, or expected value, of the sum ad product of Z, γ, ad γ P usig the Erdős Réyi radom graph G(, 1) (whose complemet is also a radom graph with edge probability 1 ). Let 2 2 G = G(, 1). 2 Let tw(h) deote the tree-width of a graph H. It is well kow that tw(h) Z(H) for all graphs of order. Sice tw(g) = o() [17], Z(G) = o(). Thus Z(G) + Z(G) = 2 o() ad Z(G) Z(G) = 2 o( 2 ), ad this establishes the the electroic joural of combiatorics 22 (2015), #P00 15

16 upper boud listed i Table 1 for coected graphs G ad G, because G ad G are both coected with probability approachig 1 as. For ay fixed ɛ > 0, γ(g) (1+ɛ) log 2 with probability goig to 1 as [15, 20]. Sice γ P (H) γ(h) for all graphs H, γ P (G) + γ P (G) 2(1 + ɛ) log 2 << + 1 as for G = G(, 1). 2 Ackowledgemets This research was supported by the America Istitute of Mathematics (AIM), the Istitute for Computatioal ad Experimetal Research i Mathematics (ICERM), ad the Natioal Sciece Foudatio (NSF) through DMS , ad the authors thak AIM, ICERM, ad NSF. We also thak S. Arumugam for sharig a paper with us via that was very helpful, Bria Wissma for fruitful discussios about prelimiary work ad for providig the Sage code for power domiatio, ad the referee for helpful commets that improved the expositio. Refereces [1] AIM Miimum Rak Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. M. Cioabă, D. Cvetković, S. M. Fallat, C. Godsil, W. Haemers, L. Hogbe, R. Mikkelso, S. Naraya, O. Pryporova, I. Sciriha, W. So, D. Stevaović, H. va der Holst, K. Vader Meule, ad A. Wagsess). Zero forcig sets ad the miimum rak of graphs. Liear Algebra App., 28: , [2] M. Aouchiche ad P. Hase. A survey of Nordhaus Gaddum type relatios. Discrete Appl. Math., 161:66 56, 201. [] S. Arumugam ad J. Paulraj Joseph. Domiatio i graphs. Iterat. J. Maagemet Systems, 11: , [] R.C. Brigham, P.Z. Chi, ad R.D. Dutto. Vertex domiatio-critical graphs. Networks, 18:17 179, [5] D. Burgarth ad V. Giovaetti. Full cotrol by locally iduced relaxatio. Phys. Rev. Lett. PRL 99, , [6] P. Dorbec, M. Heig, C. Löwestei, M. Motassier, ad A. Raspaud. Geeralized Power Domiatio i Regular Graphs. SIAM J. Discrete Math., 27: , 201. [7] J. Ekstrad, C. Erickso, H.T. Hall, D. Hay, L. Hogbe, R. Johso, N. Kigsley, S. Osbore, T. Peters, J. Roat, A. Ross, D.D. Row, N. Warberg, ad M. Youg. Positive semidefiite zero forcig. Liear Algebra Appl., 9: , 201. [8] W. Goddard ad M.A. Heig. Domiatio i plaar graphs with small diameter. J. Graph Theory, 0:1 25, [9] T.W. Hayes, S.M. Hedetiemi, S.T. Hedetiemi, ad M.A. Heig. Domiatio i graphs applied to electric power etworks. SIAM J. Discrete Math., 15: , the electroic joural of combiatorics 22 (2015), #P00 16

17 [10] T.W. Hayes, S.T. Hedetiemi, ad S. J. Slater. Fudametals of domiatio i graphs. CRC Press, Boca Rato, [11] A. Hellwig ad L. Volkma. Some upper bouds for the domiatio umber. J. Combi. Math. Combi. Comput., 57: , [12] L. Hogbe ad B. Wissma. Computatios i Sage for Nordhaus-Gaddum problems for power domiatio. PDF available at lhogbe/ng_powerdomiatio_sage.pdf. Sage worksheet available at orio.math.iastate.edu/lhogbe/ng_powerdomiatio.sws. [1] F. Jaeger ad C. Paya. Relatios du type Nordhaus-Gaddum pour le ombre d absorptio d u graphe simple. C. R. Acad. Sci. Paris Ser. A, 27:728 70, [1] D. Meierlig ad L. Volkma. Upper bouds for the domiatio umber i graphs of diameter two. Util. Math., 9: , 201. [15] S. Nikoletseas ad P. Spirakis. Near optimal domiatig sets i dese radom graphs with polyomial expected time. I J. va Leeuwe, ed., Graph Theoretic Cocepts i Computer Sciece, Lecture Notes i Computer Sciece 790, pp. 1 10, Spriger Verlag, Berli, 199. [16] E.A. Nordhaus ad J. Gaddum. O complemetary graphs. Amer. Math. Mothly, 6: , [17] G. Perarau ad O. Serra. O the tree-depth of radom graphs. Discrete Appl. Math., 168: , 201. [18] R. C. Read ad R. J. Wilso. A Atlas of Graphs. Oxford Uiversity Press, Oxford, [19] Y. Wag ad Q. Li. Super-edge-coectivity properties of graphs with diameter 2. J. Shaghai Jiaotog Uiv., (Chiese, Eglish summary), :66 69, [20] B. Wielad ad A.P. Godbole. O the domiatio umber of a radom graph. Electro. J. Comb. 8: Research Paper 7 (1 pp.), [21] M. Zhao, L. Kag ad G.J. Chag. Power domiatio i graphs. Discrete Math., 06: , the electroic joural of combiatorics 22 (2015), #P00 17