A method for eternally dominating strong grids
Alizée Gagnon, Alexander Hassler, Jerry Huang, Aaron Krim-Yee, Fionn Mc Inerney, Andrés Zacarías, Ben Seamone, Virgélot Virgile
DDiscrete Mathematics and Theoretical Computer Science
DMTCS vol. :1, 2020, A method for eternally dominating stronggrids
Aliz´ee Gagnon Alexander Hassler Jerry Huang Aaron Krim-Yee Fionn Mc Inerney Andr´es Mej´ıa Zacar´ıas Ben Seamone , ∗ Virg´elot Virgile D´epartement d’informatique et de recherche op´erationnelle, Universit´e de Montr´eal, Montreal, QC, Canada Facult´e de m´edecine et m´edecine dentaire, Universit´e catholique de Louvain Bruxelles-Woluwe, Brussels, Bel-gium David Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada Department of Bioengineering, McGill University, Montreal, QC, Canada Universit´e Cˆote d’Azur, Inria, CNRS, I3S, France Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, M´exico City, M´exico Mathematics Department, Dawson College, Montreal, QC, Canada Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada received 4 th Feb. 2019 , revised 11 th Feb. 2020 , accepted 12 th Feb. 2020 . In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard tothe attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard mayoccupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number ofguards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of theeternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small gridssuch as × n , × n , × n , and × n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] thatthe eternal domination number of these grids in general is within O ( m + n ) of their domination number which lowerbounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of stronggrids is upper bounded by mn + O ( m + n ) . We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] toprove that the eternal domination number of strong grids is upper bounded by mn + O ( m + n ) . While this does notimprove upon a recently announced bound of (cid:100) m (cid:101)(cid:100) n (cid:101) + O ( m √ n ) [Mc Inerney, Nisse, P´erennes, CIAC 2019] inthe general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is atmost . Keywords:
Eternal Domination, Combinatorial Games, Graphs, Graph Protection ∗ Corresponding author ([email protected])ISSN 1365–8050 c (cid:13) a r X i v : . [ c s . D M ] M a r Gagnon, Hassler, Huang, Krim-Yee, Mc Inerney, Mej´ıa Zacar´ıas, Seamone, Virgile
The graph security model of eternal domination was introduced in the 1990’s with the study of themilitary strategy of Emperor Constantine for defending the Roman Empire in a mathematical setting[AF95, Rev97, RR00, Ste99]. The problem which is studied in these papers, roughly put, is how to de-fend a network of cities with a limited number of armies at your disposal in such a way that an armycan always move to defend against an attack by invaders and do so for any sequence of attacks. In theoriginal version of eternal domination (also called “infinite order domination” [BCG +
04] and “eternalsecurity”[GHH05] in earlier works), k guards are placed on the vertices of a graph G so that they forma dominating set. An infinite sequence of vertices is then revealed one at a time (called “attacks”). Aftereach attack, a single guard is allowed to move to the attacked vertex. If, after each attack, the guardsmaintain a dominating set, then we say that k guards eternally dominate G . The minimum k for which k guards can eternally guard G for any sequence of attacks is called the eternal domination number of G ,and is denoted γ ∞ ( G ) .A subsequently introduced model, and the one we study here, allows any number of guards to moveon their turn. The minimum number of guards required to eternally dominate a graph G in this model(called the “all-guards move” model) is denoted γ ∞ all ( G ) , and is called the m -eternal domination numberof G . Typically, one requires that no two guards occupy the same vertex. If one allows more one guardto occupy a vertex at a given time, then the corresponding parameter typically appears in the literature as γ ∗∞ all ( G ) ; we do not consider this model here. For more variants and a background on results related toeternal domination, the reader is referred to [KM16]. We also point out that eternal domination can alsobe considered a special case of the Spy Game, where an attacker (spy) moves at speed s on the graph,while the guards are said to “control” the spy if one is distance at most d from the spy at the end of theirturn (see e.g. , [CMM +
18, CMINP20]). Eternal domination is then the special case of the Spy Game with s = diam( G ) and d = 0 . As mentioned, we consider only the “all guards move” model. The cases of paths and cycles for thisvariant of the game are trivial. In [KM09], a linear-time algorithm is given to determine γ ∞ all ( T ) forall trees T . In [BSL15], the eternal domination game was solved for proper interval graphs. In recentyears, significant effort has been made in an attempt to determine the eternal domination number ofCartesian grids, γ ∞ all ( P n (cid:3) P m ) (see Figure 1). Exact values were determined for × n Cartesian gridsin [FMvB15, GKM13] and × n Cartesian grids in [BFM13]. Bounds for × n Cartesian grids wereobtained in [FMvB15] and improved in [DM17], and exactly values for all n were recently providedin [FvB20]. Bounds for × n Cartesian grids were given in [vBvB16]. For general m × n Cartesiangrids, it is clear that γ ∞ all ( P n (cid:3) P m ) must be at least the domination number, γ ( P n (cid:3) P m ) , and so by theresult in [GPRT11] it follows that γ ∞ all ( P n (cid:3) P m ) ≥ (cid:98) ( n − m − (cid:99) − . The best known upper bound for γ ∞ all ( P n (cid:3) P m ) was determined in [LMS19], where it was shown that γ ∞ all ( P n (cid:3) P m ) ≤ mn + O ( m + n ) ,thus showing that γ ∞ all is within O ( m + n ) of the domination number.Recently, Finbow et al. studied the eternal domination game on strong grids, P n (cid:2) P m , which are,roughly, Cartesian grids where the diagonal edges exist (also known as “king” graphs)(see Figure 1).They obtained an upper bound of mn + O ( m + n ) for the eternal domination number of P n (cid:2) P m method for eternally dominating strong grids P (cid:3) P (left) and strong grid P (cid:2) P (right).[CWCF + ]. Note that it is trivially known that γ ( P n (cid:2) P m ) = (cid:100) m (cid:101)(cid:100) n (cid:101) . During the preparation of thispaper, a parallel work announced the following general lower and upper bounds of (cid:98) n (cid:99)(cid:98) m (cid:99) +Ω( n + m ) ≤ γ ∞ all ( P n (cid:2) P m ) ≤ (cid:100) m (cid:101)(cid:100) n (cid:101) + O ( m √ n ) , where n ≤ m , and thus showing, for large enough values of n and m , that γ ∞ all ( P n (cid:2) P m ) ≈ γ ( P n (cid:2) P m ) (up to low order terms) [MINP19]. We show that γ ∞ all ( P n (cid:2) P m ) ≤ mn + O ( m + n ) for all integers n, m ≥ by adapting the techniquesused in [LMS19]. In Section 2, we establish the basic strategy used in the proofs which follow. It canloosely be thought of as a strategy where the grid is partitioned into subgrids, guards which occupy thecorners of × grids stay in place, while the guards on the interior of the grid rotate in such a way that asymmetric configuration to the original is obtained. In Section 3, we show that this strategy easily worksfor the infinite Cartesian grid, and obtain the main result of the paper in Section 4. Finally, in Section5, we compare our results with those reported in [MINP19]. In the spirit of the aforementioned papersfocused on “skinny” Cartesian grids (those where the smallest dimension is bounded by or is equal tosome constant), we show that the strategy presented here gives a better upper bound for γ ∞ all ( P n (cid:2) P m ) in the case where n ≤ m and n is at most some constant. We also believe that the strategy presented isinteresting in its own right and could provide a path for analysis of strong grids in higher dimensions. Alternating strategy
We begin by formally defining the graph P n (cid:2) P m . Let V ( P n ) = { , . . . , n − } and V ( P m ) = { , . . . , m − } . Then, each vertex in P n (cid:2) P m is an ordered pair ( i, j ) , and two vertices ( i , j ) and ( i , j ) are adjacent if and only if max {| i − i | , | j − i j |} = 1 (see Figure 1).In order to eternally dominate P n (cid:2) P m , we consider a strategy that cycles through two families ofdominating sets, D and D (cid:48) (see Figure 2). Let D be a set of vertices in P n (cid:2) P m with the propertythat if ( i, j ) is in D then so are ( i + 2 , j + 1) and ( i − , j + 3) . This definition implies that D hasa periodic nature, where every seventh vertex in a row or column of P n (cid:2) P m contains a vertex in D .Hence, D can be viewed as a dominating set that contains the vertices ( i + 2 k + 7 l, j + k + 7 l ) and ( i + k + 7 l, j − k + 7 l ) for some i, j ≥ and all integers k, l such that the resulting vertices have an x -coordinate and a y -coordinate greater than or equal to zero. Similarly, D (cid:48) is the dominating set thatcontains the vertices ( i + k + 7 l, j + 3 k + 7 l ) and ( i + 2 k + 7 l, j − k + 7 l ) for some i, j ≥ and allintegers k, l such that the resulting vertices have an x -coordinate and a y -coordinate greater than or equalto zero.If the guards are in a D configuration, then the strategy for the guards is to have one guard move to theattacked vertex and for the rest of the guards to move accordingly to move into a D (cid:48) configuration and Gagnon, Hassler, Huang, Krim-Yee, Mc Inerney, Mej´ıa Zacar´ıas, Seamone, Virgile vice versa. For most attacks on the interior of the grid, only one response is possible. However, if theguards occupy a D or D (cid:48) configuration that contains ( i, j ) , then ( i − , j ) and ( i + 1 , j ) are adjacent to twoguards (assuming ( i, j ) is far enough from the borders of the grid). In the case where one of these verticesis attacked, the guard that is diagonally adjacent will defend against the attack (not the guard at ( i, j ) ).Due to the guards alternating between two families of configurations D and D (cid:48) , we call this strategy, the Alternating strategy.In the
Alternating strategy, there are anchor guards which do not move from their vertices after anattack and they are determined by which vertex is attacked and the current configuration of the guards.Essentially, the anchor guards occupy the corners of × subgrids inside which the other guards moveto protect against attacks and alternate to the next configuration. D D (cid:48)
Fig. 2: Snapshot of a × subgrid of a much larger grid, showing the positions of theguards in a D (left) and a D (cid:48) (right) configuration. P ∞ (cid:2) P ∞ Theorem 1.
The Alternating strategy eternally dominates P ∞ (cid:2) P ∞ . Proof:
Consider the guards initially beginning in a D configuration (where now coordinates are permittedto be any integer). We will show how the guards can move to a D (cid:48) configuration containing the attackedvertex for all possible attacks (within symmetry). We omit the proof of the movements of the guardsfrom a D (cid:48) configuration to a D configuration that contains the attacked vertex as it is analogous to themovements in the opposite direction.Due to symmetry, we only have to analyse the eight possible attacks on the vertices adjacent to a guardthat occupies ( i, j ) . We only consider the movements of the guards in the corresponding × subgrid ofthe attacked vertex as the remaining subgrids will all be symmetric to this one and so, the movements ofthe guards as well. Finally, we only have to analyze four of the eight possible attacks since an attack at ( i + 1 , j ) is symmetric to an attack at ( i − , j − , an attack at ( i + 1 , j + 1) is symmetric to an attack at ( i − , j ) , an attack at ( i − , j + 1) is symmetric to an attack at ( i + 1 , j − , and an attack at ( i, j + 1) is symmetric to an attack at ( i, j − . method for eternally dominating strong grids Attacked vertex Anchor vertices Guard movements ( i, j −
1) ( i, j ) → ( i, j − i − , j + 3) ( i + 1 , j − → ( i + 2 , j − i − , j −
4) ( i + 3 , j − → ( i + 4 , j − i + 6 , j + 3) ( i + 5 , j − → ( i + 5 , j )( i + 6 , j −
4) ( i + 4 , j + 2) → ( i + 3 , j + 1)( i + 2 , j + 1) → ( i + 1 , j + 2)( i + 1 , j −
1) ( i, j ) → ( i + 1 , j − i − , j + 5) ( i + 2 , j + 1) → ( i + 2 , j + 2)( i − , j −
2) ( i + 1 , j + 4) → ( i, j + 3)( i + 3 , j + 5) ( i − , j + 3) → ( i − , j + 4)( i + 3 , j −
2) ( i − , j + 2) → ( i − , j + 1)( i − , j − → ( i − , j )( i + 1 , j ) ( i, j ) → ( i − , j + 1)( i − , j + 2) ( i + 2 , j + 1) → ( i + 1 , j )( i − , j −
5) ( i + 3 , j − → ( i + 3 , j − i + 4 , j + 2) ( i + 1 , j − → ( i + 2 , j − i + 4 , j −
5) ( i − , j − → ( i, j − i − , j − → ( i − , j − i + 1 , j + 1) ( i, j ) → ( i + 1 , j + 1)( i − , j −
1) ( i + 2 , j + 1) → ( i + 3 , j )( i − , j + 6) ( i + 4 , j + 2) → ( i + 4 , j + 3)( i + 5 , j −
1) ( i + 3 , j + 5) → ( i + 2 , j + 4)( i + 5 , j + 6) ( i + 1 , j + 4) → ( i, j + 5)( i − , j + 3) → ( i − , j + 2) Tab. 1: Movements of guards from a D to a D (cid:48) configuration in the × subgrid corre-sponding to all possible attacks (less symmetric cases).It is easy to verify that the guards’ movements are possible and that they transition into a D (cid:48) config-uration after each attack (see Figure 3). Since the grid is infinite, there are an infinite number of guardsoccupying the vertices of a D or D (cid:48) configuration and so, any time a guard is required to move to a vertexby the Alternating strategy, he will always exist and, from Table 1 and Figure 3, we know the guards willalways transition from D to D (cid:48) or vice versa with the attacked vertex occupied. Thus, the guards canclearly do this strategy indefinitely and hence, they eternally dominate P ∞ (cid:2) P ∞ . P n (cid:2) P m We proceed to the case where the grid is finite and show that for n, m ≥ , γ ∞ all ( P m (cid:2) P n ) ≤ mn + O ( m + n ) . In order to facilitate obtaining an exact value for the O ( m + n ) term, we consider differentcases which depend on the divisibility of n and m . We first provide a strategy for the finite grid P n (cid:2) P m when n ≡ m ≡ and n, m ≥ , which utilizes the Alternating strategy with an adjustment todeal with the borders of the grid. We then generalize this strategy to any n × m grid for n, m ≥ byemploying two disjoint strategies. Gagnon, Hassler, Huang, Krim-Yee, Mc Inerney, Mej´ıa Zacar´ıas, Seamone, Virgile ( i, j − attacked ( i + 1 , j − attacked ( i + 1 , j ) attacked ( i + 1 , j + 1) attackedFig. 3: Movements of guards from a D to a D (cid:48) configuration in the × subgrid corre-sponding to all possible attacks (less symmetric cases). The black guard occupies vertex ( i, j ) and the four anchor guards are the guards in the corners. Theorem 2.
For any two integers n, m ≥ such that n ≡ m ≡ , γ ∞ all ( P m (cid:2) P n ) ≤ mn + ( m + n − . Proof:
We use the fact that n ≡ m ≡ to reduce the analysis of the guards’ strategy to the caseof a × grid. Essentially, the non-border vertices can be partitioned into ( n − m − × subgridssince n ≡ m ≡ . We place one guard in each of the corners of the n × m grid and these guardsnever move. Finally, we can partition the sides of the grid (not including the corners) into paths of sevenvertices.We implement the Alternating strategy in all of the × subgrids which means they will all haveidentical configurations. Hence, we can focus just on the case of the × subgrids that touch the bordervertices of the grid to ensure that the guards from the border can move into these grids when needed.Thus, we contract the n × m grid into a × grid and show a winning strategy for the guards therewhich ensures the borders of the n × m grid will be protected symmetrically for each of the paths of sevenvertices that make up the borders and that the × subgrids adjacent to the borders are symmetric to all method for eternally dominating strong grids × subgrids. This strategy can then be easily “translated” to any of the × subgrids thattouch the border vertices to gain a global strategy.We show a winning strategy for the guards in the × grid where four guards remain in the cornersindefinitely, five guards occupy each of the paths of seven vertices in between the corners (on the bordersof the grid), and seven guards from the Alternating strategy occupy the × subgrid in the middle (seeFigure 4). The five guards on each of the paths of seven vertices initially occupy the five central vertices,leaving the leaves empty. If any border vertices get attacked, then they must be one of the leaves of thepaths of seven vertices and the closest guard on the corresponding path moves to the attacked vertex. Theremaining four guards on the same path stay still, as well as all seven of the guards in the interior ofthe grid, and the guards on each of the other paths move to a symmetric formation as the path that wasattacked. Any subsequent attack on a border vertex is dealt with in the same fashion, i.e. , if the other leafis attacked, then the guards on the path move into a symmetric formation with one guard on the attackedleaf and the other four guards occupying a sequence of four vertices non-adjacent to the fifth guard andnot including any leaf. If an attack occurs on a non-leaf vertex of the path, then the five guards move backinto their initial formation which includes neither of the leaves. Note that the interior guards never moveif a border vertex is attacked and the guards on each of the paths are in symmetric positions.Now, for each guard the Alternating strategy requires to move in from a border vertex, it requires aguard to move out from the interior vertices. The exchange is easy to facilitate since the guard movingout of the interior will always move onto the same border path that the guard moving in to the interiorpreviously occupied. In all three of the possible configurations of the guards on the border vertices, theguards occupy a dominating set of the row or column of vertices adjacent to them in the × subgrid.Hence, there is a guard available to move to whichever vertex requires a guard to move to it and the guardleaving the interior can always move onto the border path as the guards can easily maneuver to leave anadjacent vertex empty for him while maintaining one of the three formations.Thus, the Alternating strategy with the extra guards on the borders of the grid, eternally dominates P m (cid:2) P n . This strategy uses ( n − m − + (2) ( m − n −
2) + 4 = mn + ( m + n − guardswhich gives our result.Fig. 4: One possible configuration of the guards in the × strong grid with the guardsin white in the corners never moving, the guards on the paths of length seven between thecorners in black, and the remaining guards in gray. Gagnon, Hassler, Huang, Krim-Yee, Mc Inerney, Mej´ıa Zacar´ıas, Seamone, Virgile
We now use Theorem 2 to prove γ ∞ all ( P m (cid:2) P n ) ≤ mn + O ( m + n ) for grids in general when n, m ≥ and to give exact values of O ( m + n ) in these bounds by employing two disjoint strategies as follows. Thestrategy from Theorem 2 is used for the largest a × b subgrid in the n × m grid such that a ≡ b ≡ and a separate strategy is used for the remaining unguarded vertices where none of the guards from thetwo strategies are ever utilised in the other strategy and never occupy a vertex that the other strategy isresponsible for protecting. Theorem 3.
For any two integers n, m ≥ , γ ∞ all ( P m (cid:2) P n ) ≤ ab + ( a + b −
1) + α (cid:100) n (cid:101) + β (cid:100) m (cid:101) − αβ where a ≡ b ≡ , ≤ n − a ≤ , ≤ m − b ≤ , and α = if m mod 7 = 21 if m mod 7 ∈ { , } if m mod 7 ∈ { , } if m mod 7 ∈ { , } β = if n mod 7 = 21 if n mod 7 ∈ { , } if n mod 7 ∈ { , } if n mod 7 ∈ { , } Proof:
The ab + ( a + b − guards follow the strategy in the proof of Theorem 2 in the a × b subgridwhich will include at least one corner of the n × m grid. Separately, there remain n − a columns and m − b rows to protect which are all found on the same side of the n × m grid due to the placement ofthe a × b grid. That is, there are n − a consecutive remaining columns and m − b consecutive remainingrows which overlap near one corner of the n × m grid (see Figure 5). We can guard the m − b remainingrows with α (cid:100) n (cid:101) guards, since one guard every two vertices can protect two rows since the two rows arepartitioned into disjoint K (plus some remainder due to divisibility) and one guard is assigned to each K which clearly he can protect. Thus, we use (cid:100) n (cid:101) guards for every two rows that remain and if there arean odd number of rows remaining, then we use (cid:100) n (cid:101) guards to protect the last remaining row. Similarly, β (cid:100) m (cid:101) corresponds to the number of guards required to protect the n − a remaining columns.Since wehave over-counted by αβ overlapping guards, the bound follows.Fig. 5: The n × m strong grid with the area in white representing the a × b subgrid and thearea in gray representing the remaining rows and columns. method for eternally dominating strong grids Corollary 4.
For any two integers n, m ≥ , γ ∞ all ( P m (cid:2) P n ) ≤ mn + O ( m + n ) . Proof:
This follows directly from Theorem 3.
In a parallel work, Mc Inerney, Nisse, and P´erennes [MINP19] show that if m ≥ n , then γ ∞ all ( P n (cid:2) P m ) ≤(cid:100) m (cid:101)(cid:100) n (cid:101) + O ( m √ n ) . The general configuration which is maintained is to (a) fill some number of rowsand columns with stationary guards so that the dimensions of the remaining m ∗ × n ∗ grid satisfy necessarydivisibility conditions, (b) add additional rows and columns of guards to allow passage of guards aroundthe outside of the subgrid, (c) partition the subgrid into m ∗ × k smaller subgrids, (d) place guards alongboundary layers of each subgrid, and (e) place one guard for every nine vertices of the interior of eachsubgrid in such a way that every attack has a response, transferring guards as need be through the boundarylayers.In the best case ( i.e. , it is not necessary to fill some rows/columns with guards to ensure the remainingsubgrid satisfies divisibility conditions), the bound from [MINP19] is (cid:18) m − α α − (cid:19) (cid:18)(cid:24) ( n − α − (cid:25) + 2 n + 6 α − (cid:19) + 2 α ( m + n − α ) . (1)where α = k − + 1 and k is the greatest integer less or equal to √ n for which k ≡ (note that k > √ n − ).The worst case for our bound is when α = β = 3 , a = n − , and b = m − , as this requires packingthe most stationary guards around two sides of the grid. This gives ( m − n − m + n − (cid:108) m (cid:109) + 3 (cid:108) n (cid:109) − . (2)Fig. 6: Comparison of bounds (1) and (2); m as horizontal axis, n as vertical axis0 Gagnon, Hassler, Huang, Krim-Yee, Mc Inerney, Mej´ıa Zacar´ıas, Seamone, Virgile
The dark shaded region shown in the graph in Figure 6 gives the values of m and n for which the boundin (2) bests the bound in (1), using k = √ n − to express (1) as a function of m and n only. Note thatthe bounding function as m → ∞ eventually stays strictly between n = 6179 and n = 6180 , and thusour result is best when n < .We point out that the authors of [MINP19] did not attempt to optimize the constants in their result(nor did we in this paper), only to show that the domination number plus some low order terms was anupper bound for γ ∞ all . However, the “dense” guards surrounding each subgrid is an integral part of theirargument, leading to the O ( m √ n ) term in their result which cannot be dropped (unless a new method isfound to guard the boundaries). As a result, even with optimization of constants, our strategy should bepreferred for sufficiently “skinny” n × m strong grids. This work was undertaken while the second, third, and fourth authors were affiliated with Dawson Col-lege,and while the eighth author was affiliated with Universit´e de Montr´eal. The sixth author acknowl-edges the generous support of MITACS and the Globalink Program. The fourth and seventh authorsreceived financial support for this research from the Fonds de recherche du Qu´ebec - Nature et technolo-gies.
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