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The game total domination problem is log-complete in PSPACE
Accepted Manuscript The Game Total Domination Problem is Log-Complete in PSPACE
Boštjan Brešar, Michael A. Henning
PII: DOI: Reference:
S0020-0190(17)30099-6 http://dx.doi.org/10.1016/j.ipl.2017.05.007 IPL 5541
To appear in:
Information Processing Letters
Received date: Revised date: Accepted date:
28 September 2016 22 May 2017 25 May 2017
Please cite this article in press as: B. Brešar, M.A. Henning, The Game Total Domination Problem is Log-Complete in PSPACE, Inf. Process. Lett. (2017), http://dx.doi.org/10.1016/j.ipl.2017.05.007
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Highlights • We prove that verifying whether the game total domination number of a graph is bounded by a given integer is log-complete in PSPACE. • We present the key arguments for the PSPACE-completeness of the Staller-start version of the total domination game. • The translation of the POS-CNF problem to a newly established PSACE-complete problem.
The Game Total Domination Problem is Log-Complete in PSPACE 1,2
Boˇstjan Breˇsar and 3 Michael A. Henning
1
Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia 2 Institute of Mathematics, Physics and Mechanics, Slovenia [email protected] 3
Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006 South Africa [email protected]
Abstract In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462]. A vertex totally dominates another vertex in a graph G if they are neighbors. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game played on G consists of two players, named Dominator and Staller, who alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. The game ends when the set of vertices chosen becomes a total dominating set in G. Dominator wishes to end the game with a minimum number of vertices chosen, and Staller wishes to end the game with as many vertices chosen as possible. The game total domination number of G is the number of vertices chosen when Dominator starts the game and both players play optimally. In this paper, we show that verifying whether the game total domination number of a graph is bounded by a given integer is log-complete in PSPACE. Keywords: Total domination game; Computational complexity; PSPACE-complete problems; POS-CNF problem. AMS subject classification: 05C57, 91A43, 05C69
1
Introduction
In this paper, we continue the study of the domination game in graphs introduced in [3] and extensively studied afterward in [1, 3, 4, 7, 13, 14] and elsewhere. The total version of the domination game was recently investigated in [10], where it was shown, among other results, that the two versions differ significantly (in particular, there exists a family of graphs Gn whose game domination number is 2n − 1, while the game total domination number is n + 1). A vertex u in a graph G totally dominates a vertex v if u is adjacent to v in G. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game consists of two players called Dominator and Staller, who take turns choosing a vertex from G. Each vertex chosen must totally dominate at least one vertex not totally 1
dominated by the set of vertices previously chosen. We call such a chosen vertex a playable vertex ; that is, at a particular point in the game, a playable vertex is a vertex that is adjacent to a vertex that is not totally dominated by a vertex previously chosen. We also call a playable vertex a legal move. A vertex is unplayable if it is not a legal move. The game ends when G has no playable vertices in which case the set of vertices chosen is necessarily a total dominating set in G. Dominator wishes to end the game with a minimum number of vertices chosen, and Staller wishes to end the game with as many vertices chosen as possible. The game total domination number, γtg (G), of G is the minimum possible number of vertices chosen when Dominator starts the game and both players play according to the rules. The Staller (G), of G is the number of vertices chosen when Staller start game total domination number, γtg starts the game and both players play optimally. We refer to the Dominator-start total domination game and the Staller-start total domination game simply as the Dominator-start game and Staller-start game, respectively. We denote the sequence of moves played in the Dominator-start game by d1 , s1 , d2 , s2 , . . ., where di and si are the ith moves played by Dominator and Staller, respectively, where i ≥ 1. Further, we denote the sequence of moves played in the Staller-start game by s1 , d1 , s2 , d2 , . . ., where si and di are the ith moves played by Staller and Dominator, respectively, where i ≥ 1. Adopting the terminology in [13], a partially-dominated graph is a graph in which we suppose that some vertices have already been totally dominated, and need not be totally dominated again to complete the game. Given a graph G and a subset S of vertices of G, we denote by G|S the partially-dominated graph in which the vertices of S in G are already totally dominated. We use γtg (G|S) (resp. γtg (G|S)) to denote the number of turns remaining in the game on G|S under optimal play when Dominator (resp. Staller) has the next turn. In [10], the authors present a key lemma, named the Total Continuation Principle, which in particular implies that the number of moves in the Dominator-start game and the Staller-start differ by at most 1; that is, |γtg (G) − γtg (G)| ≤ 1 for every graph G (with no isolated vertex). Lemma 1 ([10]) (Total Continuation Principle) Let G be a graph, and let A, B ⊆ V (G). If B ⊆ A, (G|A) ≤ γtg (G|B). then γtg (G|A) ≤ γtg (G|B) and γtg (G) is a challenging problem, and is currently only Determining the exact value of γtg (G) and γtg known for paths and cycles [9]. A bound on the game total domination number for general graphs is established in [11] where it is shown that if G is a graph on n vertices in which every component (G) ≤ (4n + 2)/5. This upper bound is contains at least three vertices, then γtg (G) ≤ 54 n and γtg 3 improved in [12] to γtg (G) ≤ 4 n over the class of graphs with the degree sum of adjacent vertices at least 4 and with no two vertices of degree 1 at distance 4 apart in G. Further, the upper bound 8 n over the class of graphs with minimum degree at least 2. is improved in [6] to γtg (G) ≤ 11 In this paper, we study the algorithmic complexity of determining the game total domination number of a given graph. The algorithmic approach to this problem is not at all likely to be polynomial in time. Even worse, it is questionable whether it is NP-hard. It was proven in [2] that the complexity of verifying whether the game domination number of a graph is bounded by a given integer is in the class of PSPACE-complete problems. Hence the decision version of the game domination problem is computationally harder than any NP-complete problem, unless NP=PSPACE. In this paper we use the same known PSPACE-complete problem (notably the POS-CNF problem; see [15] for the complexity result on this problem) to verify that the complexity of the decision version of the total game domination number is PSPACE-complete as well. Formally, we are dealing with the following Game Total Domination Problem:
Game Total Domination Problem Input: A graph G, and an integer m. Question: Is γtg (G) ≤ m? 2
We establish the following result, a proof of which is given in the next section. Theorem 2 Game Total Domination Problem is PSPACE-complete. More precisely, since the reduction from the POS-CNF problem to the game total domination problem can be computed with a working space which is logarithmic in size of a formula in the POS-CNF problem, and since the POS-CNF is log-complete in PSPACE, we derive that Game Total Domination Problem is also log-complete in PSPACE.
2
A Proof of Theorem 2
We present the reduction to the Game Total Domination Problem from the POS-CNF problem, which is known to be log-complete in PSPACE [15]. In POS-CNF we are given a set of variables, and a formula that is a conjunction of disjunctive clauses, in which there is no negations of variables. Two players alternate turns, the first player setting a previously unset variable TRUE, and the second player setting such a variable FALSE. After all variables are set, the first player wins if the formula is TRUE, otherwise the second player wins. In what follows, we use the standard notation [k] = {1, . . . , k}. Let T be the gadget graph shown in Figure 1, where V (T ) = {a, b, b , z, z , x, y, w}. b
x z
y w z
a
b
Figure 1: The gadget, T , used in the construction, representing a variable The following observation summarizes the properties of the graph T needed in the proof of our main result. Note that the rules of the game do not allow any of the players passing a move. However, the graph GF that appears in our main construction consist of several copies of the gadget graph T , and it may happen during the game played on GF that after some moves were played in a subgraph Ti isomorphic to the gadget graph T that one of the players chooses to make a move outside Ti before all vertices of this subgraph are totally dominated. In this case, the other player could be the first one to play again in the subgraph Ti . From the point of view of the graph T , this corresponds to one player passing a move in T . Observation 1 Let T be the gadget graph shown in Figure 1. The following properties hold in T . (a) γtg (T ) = 3, γtg (T |a) = 3, γtg (T ) = 3, γtg (T |a) = 3.
(b) If Dominator starts the game on T with d1 = a, then Staller can ensure three moves are played in T by playing s1 = b. In this case, Dominator can finish the game on T in three steps by playing d2 = x; if, however, Dominator passes his second move, then Staller can ensure four moves are played in T by playing the vertex b .
3
(c) If Dominator starts the game on T with d1 = a, and Staller passes her first move, then Dominator can finish the game on T in only two steps by playing d2 = x (or d2 = w). (d) If Staller starts the game on T with s1 = b, then Dominator can ensure three moves are played in T by playing d1 = x. (e) If Staller starts the game on T with s1 = b, and Dominator responds by playing d1 = a, then Staller can enforce four moves on T by playing s2 = b . (f) If Staller starts the game on T with s1 = b, and Dominator passes his first move, then Staller can enforce four moves on T by playing s2 = b . (g) If Staller starts the game on T with s1 ∈ / {b, b }, then Dominator can finish the game on T in his next move, and so only two moves are played on T . Given a formula F using k variables and n disjunctive clauses, we build a graph GF , having 8(k + 1) + n + 9 vertices, as follows. For each variable Xi (where i ∈ [k]) we add to the graph a copy, Ti , of the gadget graph, T , shown in Figure 1. For each clause Cj (where j ∈ [n]) we add a vertex cj to the graph, and make cj adjacent to a vertex ai , whenever the variable Xi appears in the clause Cj . In addition, we add to the graph a special gadget, T0 . For each vertex u ∈ V (T ) = {a, b, b , x, y, z, z , w}, we denote the associated vertex in the ith gadget Ti by ui , where i ∈ {0} ∪ [k]. We make the vertex a0 in the gadget T0 (corresponding to the vertex a in the gadget T ) adjacent to all vertices cj , for j ∈ [n]. Finally, we add a vertex v and make it adjacent to all vertices ai , where i ∈ {0} ∪ [k], and add vertices e1 , . . . , e4 , f1 , . . . , f4 , and make each vertex ei adjacent to both vertices v and fi for all i ∈ [4]. We note that the set {c1 , . . . , cn } is an independent set in GF . To illustrate the construction of the graph GF , consider the formula F = C1 ∧ C2 ∧ C3 , where C1 = X1 ∨ X3 , C2 = X2 ∨ X4 , and C3 = X3 ∨ X4 . In this case, there are k = 4 variables and n = 3 disjunctive clauses. The associated graph GF is illustrated in Figure 2, where in each gadget, Ti , where i ∈ {0} ∪ [4], we only label the vertex ai (corresponding to the vertex a in the gadget T ). f1 e1
f2 e2
f3 e3
f4 e4
v
X1
a0
a1
X2
X3
X4
a3
a2
c1
c2
a4
c3
Figure 2: The graph, GF , associated with the formula F = (X1 ∨ X3 ) ∧ (X2 ∨ X4 ) ∧ (X3 ∨ X4 ). 4
The following result implies the proof of our main theorem. We say that a gadget is opened if a move has been played in that gadget. Theorem 3 Let F be a formula with k variables, which is a conjunction of n disjunctive clauses, and let GF be the corresponding graph. Player 1 has a winning strategy for F in the POS-CNF game if and only if γtg (GF ) ≤ 3k + 8. Proof. We first assume that Player 1 has a winning strategy for F in the POS-CNF game, and describe a strategy for Dominator that guarantees that at most 3k + 8 moves will be played in the total domination game played in the graph GF . Dominator plays as his first move, d1 , the vertex v, thus totally dominating all vertices ai , where i ∈ {0} ∪ [k] and all vertices ei , where i ∈ [4]. This implies that no vertex cj , where j ∈ [n], and also no vertex fj , where j ∈ [4], is a legal move in the remaining part of the game. (Eventually, each vertex ei will have to be played in the game to totally dominate the vertex fi for all i ∈ [4]. If during the game, Staller plays the vertex ei , then on the next move of Dominator he plays a vertex ej that has not yet been played for some j = i. By this strategy, the vertices ei have no effect on the rest of the game, and we may assume that they are played at the end of the game.) We remark that whenever Staller makes a move which is the first move played from a gadget, Ti , for some i ∈ {0} ∪ [k], then, by Observation 1(g), it is in her best interest to play the vertex bi or the vertex bi . Renaming vertices, if necessary, we may assume that Staller plays the vertex bi in such circumstances. Suppose that Staller plays as her first move, s1 , a vertex from the gadget Ti , for some i ∈ {0} ∪ [k]. By our earlier assumption, s1 = bi . Dominator responds by playing as his second move, d2 , the vertex xi from the same gadget Ti . This strategy of Dominator implies that all vertices in the gadget Ti are totally dominated by the played vertices, except for exactly one vertex, namely the vertex wi . Thus, only one additional vertex can be played among the vertices in Ti , and such a vertex is necessarily a neighbor of wi (in order for the move to be legal). If Staller plays as her second move, s2 , a vertex from the gadget Ti , for some i ∈ {0} ∪ [k] \ {i}, then, by our earlier assumption, s2 = bi and Dominator responds by playing as his third move, d3 , the vertex xi from the same gadget Ti . Dominator continues with this strategy by always responding to a move bj of Staller which is the first move played in some gadget Tj , where j ≥ 1, by playing the vertex xj in the same gadget Tj . Renaming variables and clauses, if necessary, we may assume for notational convenience, that Staller’s first moves s1 , . . . , s are the vertices b1 , . . . , b , from the first distinct gadgets T1 , . . . , T , and Dominator’s first + 1 moves d1 , d2 , . . . , d+1 are the vertices v, x1 , . . . , x . Further, we assume that is the largest such integer. Possibly, = k. In view of the preceding arguments, we may therefore assume, renaming variables and clauses if necessary, that for ≥ 0, Staller’s first + 1 moves s1 , . . . , s+1 are the vertices b0 , . . . , b , from the first + 1 gadgets T0 , T1 , . . . , T , and Dominator’s first + 2 moves d1 , d2 , . . . , d+2 are the vertices v, x0 , . . . , x . Further, we may assume that is the largest such integer. Possibly, = 0 or = k. By our choice of the integer , Staller’s ( + 2)nd move is played in one of the previously opened gadgets Ti , where i = 0 or i ∈ [] and ≥ 1. As observed earlier, this move of Staller is a neighbor of wi , and all vertices of Ti are now totally dominated by the played vertices. If i = 0 (and so, ≥ 1 and i ∈ []), then Dominator responds to this move of Staller by playing as his ( + 3)rd move the vertex a0 . As before, by Observation 1(a), Dominator can make sure the game takes at most 3k + 8 moves to complete. Hence, we may assume that Staller’s ( + 2)nd move is played in the gadget T0 and is not the vertex a0 . Thus, three vertices are currently played from T0 and all vertices of T0 are totally dominated. We remark that at this stage of the game, none of the clause vertices, cj , is yet totally dominated. Dominator now plays as his ( + 3)rd move the vertex at in the gadget Tt , where Xt is the first variable Player 1 would set TRUE in the POS-CNF game. If t ∈ [], then all vertices of Tt are now totally dominated by the played vertices. If t ∈ / [], then < k and t ≥ + 1. In this case, if Staller 5
responds by playing her ( + 3)rd move in the gadget Tt , then Dominator plays the vertex xt in reply to Staller’s move, making all vertices of Tt totally dominated, and thereby ensures the game takes at most 3k + 8 moves to complete in view of the preceding arguments. Hence, Staller plays as her ( + 3)rd move either the first move played in some gadget Ts that has not previously been opened, or a third move in a previously opened gadget Tr . In the former case, Dominator responds to Staller’s move by playing the vertex xs . In this case, only the vertex ws remains totally undominated in Ts ; hence, by the Total Continuation Principle, if Dominator is going to be the last to play in Ts , he would choose as , while Staller would in such event choose some other neighbor of ws . In the latter case, that is, if Staller plays the third move in a previously opened gadget Tr , such a move of Staller is a neighbor of wr , and all vertices of Tr are now totally dominated by the played vertices. Dominator considers such a move of Staller as Player 2 setting the associated variable Xr FALSE in the POS-CNF game, and follows the POS-CNF winning strategy of Player 1. Thus, if Xp is the next variable that Player 1 would set TRUE in the POS-CNF game in response to the previous move by Player 2 which set the variable Xr FALSE, then in the total domination game played in GF , Dominator would play as his next move the vertex ap . With this strategy of Dominator, he can guarantee that at most 3k + 8 moves are played in order to complete the game. Now we assume that Player 2 has a winning strategy for F in the POS-CNF game, and describe a strategy for Staller that guarantees that at least 3k + 9 moves will be played in the total domination game played in the graph GF . The base part of Staller’s strategy is that whenever Dominator plays in some gadget Ti , her response is to reply with a move in the same gadget Ti , unless Dominator played the last (in most cases the third) move in Ti , which makes all vertices of Ti totally dominated. By Observation 1(a) and the structure of GF , this implies that at least three moves will be played in each Ti . If during the game, Dominator plays the vertex ei , then on the next move of Staller she plays a vertex ej that has not yet been played for some j = i. By this strategy, the vertices ei have no effect on the rest of the game, and we may assume as in the first part of the proof that they are played at the end of the game. First let us assume that as his first move, d1 , Dominator chooses the vertex v, and so d1 = v. Staller replies in T0 by playing the move s1 = b0 . If Dominator as the next move or at some later point in the game chooses a0 as his move, then, by Observation 1(e), Staller can enforce four moves are played in T0 by choosing b0 as her next move; this, together with her basic strategy, by which (at least) three moves will be played in each Ti , already guarantees that at least 3k + 9 moves will be played in the game (counting also four moves played on the vertices ei for i ∈ [4]). If Dominator plays some vertex in T0 different from a0 as his second move, then Staller responds by playing a legal move in T0 different from a0 . In this way, all vertices of T0 are totally dominated, and three vertices of T0 are played. Further, the vertex a0 is not one of the played vertices in T0 . Thus, we may assume that the second move of Dominator is d2 = x0 and Staller’s second move is s2 = y0 . The third move of Dominator is in Tt for some t ∈ [k], hence, noting that in the graph GF the set N (wt ) is contained in N (xt ), while N (zt ), N (zt ) and N (yt ) are all contained in N (at ), by the Total Continuation Principle we may assume that Dominator chooses one of the vertices from {at , bt , bt , xt }. By her strategy, Staller follows by choosing a vertex from Tt as her third move. If d3 = at , then Staller can ensure that after her third move the vertex wt is not totally dominated. If, on the other hand, d3 = at , Staller responds by playing s3 = bt . Following the first alternative in the previous paragraph, Dominator may open several other gadget subgraphs Ti by playing a vertex different from ai , whence Staller follows by playing another move in Ti that ensures that wi is not totally dominated after her move. Continuing in this way, suppose that + 1 gadget subgraphs are opened for some ≥ 0. Renaming variables and clauses if necessary, we may assume that T0 , T1 , . . . , T are the opened gadgets, where all vertices of T0 are totally dominated, while in Ti , for i ∈ [], the vertex wi is not yet totally dominated. Further we 6
assume that is the largest such integer, and possibly, = k. Hence, the ( + 3)rd move of Dominator is the vertex at , for some t ∈ [k]. Possibly, t ≤ in which case all vertices of the gadget subgraph Tt are totally dominated. We proceed further with the following claim. Claim A During the game, if Dominator opens two different gadgets Ti by playing vertex ai without first ensuring that all vertices in one of these two gadgets are totally dominated, then Staller guarantees that at least 3k + 9 moves will be played. Proof. Suppose that as and at are moves of Dominator and these moves open the gadgets Ts and Tt without first ensuring that all vertices in one of these two gadgets are totally dominated. The corresponding responses of Staller are bs and bt , by her basic strategy. Further assume that xs and xt have not yet been played, and not all vertices of Ts and Tt are totally dominated. Then, at some point, Dominator will have to choose a vertex from a previously opened gadget, thereby totally dominating every vertex from that gadget. Renaming indices if necessary, we may assume that among the two gadgets, Ts and Tt , Dominator first returns to Tt by playing the vertex xt as the third move played in Tt . Now, Staller responds by playing the vertex bs in the gadget Ts , which, by Observation 1(f), implies that four moves are played in Ts ; hence at least 3k + 9 moves will be played in the game, as desired. This completes the proof of the Claim A. (2) We remark that in view of our preceding arguments, it is only in Staller’s interest if Dominator opens gadgets Ti by playing a vertex different from ai . More precisely, if Dominator opens two different gadgets Ti by playing wherever he wishes in these gadgets and without first ensuring that all vertices in one of these two gadgets are totally dominated, then Staller guarantees that at least 3 moves will be played in each gadget (and if in both gadgets vertices ai were played, then Staller can enforce that at least 3k + 9 moves will be played in total). We may therefore assume that whenever Dominator opens a gadget Ti by playing ai , for some i ∈ [k], and Staller, according to her strategy, responds by playing the vertex bi as her next move, then Dominator immediately responds to Staller’s move by playing the vertex xi that belongs to the gadget Ti , thereby totally dominating every vertex from Ti . We now return to the proof of Theorem 3. Recall that the ( + 3)rd move of Dominator is the vertex at , for some t ∈ [k]. Staller considers the move at of Dominator as if Player 1 would set variable Xt TRUE. We consider the case when t ≤ and the case when t > separately. Case 1. Suppose t ≤ . In this case, all vertices of Tt are totally dominated after the move d+3 = at . Staller responds by playing in Tr , where the optimal response of Player 2 in the POSCNF game to Player 1 setting the variable Xt TRUE, is setting the variable Xr FALSE. There are two possibilities for Staller’s move. If r ≤ , then Staller plays a vertex in Tr − ar , which totally dominates wr , and all vertices of Tr are then totally dominated. On the other hand, if r > , Staller plays br . By Observation 1(e), in order to avoid four moves being played in the gadget subgraph Tr , Dominator responds to Staller’s move, br , by playing the vertex xr . Staller in turn responds to Dominator’s move by playing the vertex yr , thereby totally dominating every vertex from the gadget Tr . Case 2. Suppose t > . In this case, Dominator’s ( + 3)rd move opens the gadget Tt . Staller responds by choosing as her ( + 3)rd move the vertex bt . By our assumption following the proof of Claim A, Dominator responds in Tt by playing xt . Thus, Dominator’s ( + 4)th move plays the vertex xt , thereby totally dominating all vertices of Ti . The response of Staller, s+4 , is analogous as in Case 1. Notably, Staller responds by playing in Tr , where the optimal response of Player 2 in the POS-CNF game to Player 1 setting the variable Xt TRUE, is setting the variable Xr FALSE. As in Case 1, there are two possibilities with respect to r being bigger than or not, and they are dealt with in the same way as in Case 1. 7
In the remainder of the game, Dominator may choose a vertex at in some Tt in which case Staller responds in the same way as in one of the above cases, depending on whether Dominator opens the gadget Tt by playing at or whether Tt is already opened prior to his move. In both cases, by the same reasoning as above, Staller responds in such a way that one of the gadget subgraphs Tr becomes totally dominated without ar being played. On the other hand, Dominator may also choose to open a new gadget subgraph Tt without playing at . In this case, Staller only responds in that gadget subgraph, and it is still not determined, whether at will be played or not. (Such moves are not reflected in the POS-CNF game, which is simultaneously considered by Staller.) By Staller’s strategy, the game in GF reflects the POS-CNF game in F. We derive that after all 3(k + 1) moves are played in the gadgets Ti , i ∈ {0} ∪ [k], there is still a clause vertex cj , which is not yet totally dominated. Hence, at least one additional vertex must be played in order to make sure that the clause vertices are all totally dominated. Thus, including the first move d1 = v of Dominator, and moves on vertices e1 , e2 , e3 , e4 , at least 3k + 9 moves are played in the game. Finally, let us assume that the first move, d1 , of Dominator is not the vertex v. By the Total Continuation Principle, it is never in Dominator’s best interests to play a clause vertex cj , since in this case he can always do at least as well by playing the vertex v which totally dominates a superset of vertices in N (cj ). As discussed earlier, we may assume that the vertices ei are played at the end of the game. Indeed, if Dominator plays the vertex ei , then Staller can only benefit from this move and continue to follow the base part of her strategy as described earlier. Hence we may assume that Dominator starts to play in one of the gadgets Ti , for some i ∈ {0} ∪ [k]. Staller now plays as her first move, s1 , the vertex f1 . There are several cases with respect to the next few moves of Dominator. He could continue to play in gadget subgraphs, in which case, Staller would choose f2 , f3 and eventually also f4 . Alternatively, Dominator could in one of his first four moves choose v and prevent fi as a legal move. Let us only consider the first case where Staller plays the vertices f1 , . . . , f4 on her first four moves. We remark that the other cases are similar, only slightly shorter in argumentation. By Observation 1(c), Dominator can finish the game in the gadget subgraph Ti in only two moves, if Staller chooses not to respond to Dominator’s move in Ti . Hence, after the first five moves of Dominator the following can occur: • Five gadgets Ti are opened with only one move in each of the gadgets played by Dominator; • Four gadgets Ti are opened, where all vertices in one gadget are totally dominated, whereas in other three gadgets Dominator played only one move; • Three gadgets Ti are opened, where two gadgets have all vertices totally dominated, whereas in one gadget only one move of Dominator was played. In addition, all vertices f1 , . . . , f4 were chosen by Staller, and it is Staller’s turn to play. In her next move, s5 , Staller plays in one of the opened but not yet totally dominated gadgets, thereby ensuring at least three moves will be played in this gadget (using Observation 1(a)). Regardless of which of the three possibilities above occur as a consequence of the first five moves played by Dominator, Staller can guarantee with her strategy that in at most two gadgets are exactly two moves played, while in the remaining k − 1 gadgets at least three moves are played. In addition, all eight vertices e1 , . . . , e4 , f1 , . . . , f4 are played, implying that at least 2 × 2 + 3(k − 1) + 8 = 3k + 9 moves will be played in the game, which makes the final case of Staller’s strategy complete.
3
Concluding Remarks
In this paper we proved the PSPACE-completeness of Game Total Domination Problem by using a reduction from the POS-CNF problem, in which a formula F is given, to the total domination 8
game of a special graph GF . It is easy to see that the reduction we used can be computed with a working space of logarithmic size with respect to the entry, making this problem log-complete in PSPACE. An analogous result, i.e., PSPACE-completeness, is true also for the Staller-start version of the game total domination number. Moreover, even the same construction can be used, translating a formula F to the graph GF , and it can be proved, that Player 1 has a winning strategy for F in (GF ) ≤ 3k + 9. Let us give some brief ideas about the proof the POS-CNF game if and only if γtg of this result. It is easy to see that when Player 1 has a winning strategy for F, then Dominator can ensure at most 3k + 9 moves are played in GF , when Staller starts the game. Indeed, after any first move of Staller, the graph GF becomes partially totally dominated, and by combining the Total Continuation Principle with the proof of Theorem 3, he can ensure that only 3k + 8 moves are played after the first move of Staller. On the other hand, when Player 2 has a winning strategy for a formula F in POS-CNF, Staller’s strategy is to start in a clause vertex cj (note that there exists a clause vertex that is not adjacent to all vertices ai , hence v is still a legal move after Staller’s first move). If Dominator’s response is to play v, then the game already reduces to the previous version of the game in GF , where Staller can ensure 3k + 9 moves are played, altogether 3k + 10 moves counting her first move. However, if Dominator responds in some gadget, then the situation becomes similar to the final part of Staller’s strategy in the proof above, where Staller chooses vertices fi .
Acknowledgements We are grateful to the reviewers for their helpful remarks and suggestions. B. Breˇsar is supported in part by the Slovenian Research Agency (ARSS) under the grants P1-0297, J1-7110, and N1-0043. Research of M. A. Henning is supported in part by the South African National Research Foundation and the University of Johannesburg.
References [1] B. Breˇsar, P. Dorbec, S. Klavˇzar, and G. Koˇsmrlj, Domination game: effect of edge- and vertexremoval. Discrete Math. 330 (2014), 1–10. [2] B. Breˇsar, P. Dorbec, S. Klavˇzar, G. Koˇsmrlj, and G. Renault, Complexity of the game domination problem, Theoret. Comput. Sci. 648 (2016), 1–7. [3] B. Breˇsar, S. Klavˇzar, and D. F. Rall, Domination game and an imagination strategy. SIAM J. Discrete Math. 24 (2010), 979–991. [4] B. Breˇsar, S. Klavˇzar, D. F. Rall, Domination game played on trees and spanning subgraphs. Discrete Math. 313 (2013), 915–923. [5] B. Breˇsar, S. Klavˇzar, G. Koˇsmrlj, D. F. Rall, Domination game: extremal families of graphs for the 3/5-conjectures. Discrete Appl. Math. 161 (2013), 1308–1316. [6] C. Bujt´ as, M. A. Henning, and Z. Tuza, Transversal game on hypergraphs and the 34 -Conjecture on the total domination game. SIAM J. Discrete Math. 30(3) (2016), 1830–1847. [7] Cs. Bujt´as, On the game domination number of graphs with given minimum degree., Electron. J. Combin. 22(3) (2015), #P3.29. [8] P. Dorbec, G. Koˇsmrlj, and G. Renault, The domination game played on unions of graphs, Discrete Math. 338 (2015), 71–79. 9
[9] P. Dorbec and M. A. Henning, Game total domination for cycles and paths, Discrete Appl. Math. 208 (2016), 7–18. [10] M. A. Henning, S. Klavˇzar, and D. F. Rall, Total version of the domination game. Graphs Combin. 31(5) (2015), 1453–1462. [11] M. A. Henning, S. Klavˇzar, and D. F. Rall, Total domination game played in forest. Combinatorica 37(2) (2017), 223–251. [12] M. A. Henning, and D. F. Rall, Progress towards the total domination game Discrete Math. 339 (2016), 2620–2627.
3 4 -Conjecture.
[13] W. B. Kinnersley, D. B. West, R. Zamani, Extremal problems for game domination number, SIAM J. Discrete Math. 27 (2013), 2090–2017. [14] G. Koˇsmrlj, Realizations of the game domination number, J. Comb. Optim. 28 (2014), 447–461. [15] T. J. Schaefer, On the complexity of some two-person perfect-information games. J. Comput. System Sci. 16 (1978), 185–225.
10
Boštjan Brešar, Michael A. Henning
PII: DOI: Reference:
S0020-0190(17)30099-6 http://dx.doi.org/10.1016/j.ipl.2017.05.007 IPL 5541
To appear in:
Information Processing Letters
Received date: Revised date: Accepted date:
28 September 2016 22 May 2017 25 May 2017
Please cite this article in press as: B. Brešar, M.A. Henning, The Game Total Domination Problem is Log-Complete in PSPACE, Inf. Process. Lett. (2017), http://dx.doi.org/10.1016/j.ipl.2017.05.007
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Highlights • We prove that verifying whether the game total domination number of a graph is bounded by a given integer is log-complete in PSPACE. • We present the key arguments for the PSPACE-completeness of the Staller-start version of the total domination game. • The translation of the POS-CNF problem to a newly established PSACE-complete problem.
The Game Total Domination Problem is Log-Complete in PSPACE 1,2
Boˇstjan Breˇsar and 3 Michael A. Henning
1
Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia 2 Institute of Mathematics, Physics and Mechanics, Slovenia [email protected] 3
Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006 South Africa [email protected]
Abstract In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462]. A vertex totally dominates another vertex in a graph G if they are neighbors. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game played on G consists of two players, named Dominator and Staller, who alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. The game ends when the set of vertices chosen becomes a total dominating set in G. Dominator wishes to end the game with a minimum number of vertices chosen, and Staller wishes to end the game with as many vertices chosen as possible. The game total domination number of G is the number of vertices chosen when Dominator starts the game and both players play optimally. In this paper, we show that verifying whether the game total domination number of a graph is bounded by a given integer is log-complete in PSPACE. Keywords: Total domination game; Computational complexity; PSPACE-complete problems; POS-CNF problem. AMS subject classification: 05C57, 91A43, 05C69
1
Introduction
In this paper, we continue the study of the domination game in graphs introduced in [3] and extensively studied afterward in [1, 3, 4, 7, 13, 14] and elsewhere. The total version of the domination game was recently investigated in [10], where it was shown, among other results, that the two versions differ significantly (in particular, there exists a family of graphs Gn whose game domination number is 2n − 1, while the game total domination number is n + 1). A vertex u in a graph G totally dominates a vertex v if u is adjacent to v in G. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game consists of two players called Dominator and Staller, who take turns choosing a vertex from G. Each vertex chosen must totally dominate at least one vertex not totally 1
dominated by the set of vertices previously chosen. We call such a chosen vertex a playable vertex ; that is, at a particular point in the game, a playable vertex is a vertex that is adjacent to a vertex that is not totally dominated by a vertex previously chosen. We also call a playable vertex a legal move. A vertex is unplayable if it is not a legal move. The game ends when G has no playable vertices in which case the set of vertices chosen is necessarily a total dominating set in G. Dominator wishes to end the game with a minimum number of vertices chosen, and Staller wishes to end the game with as many vertices chosen as possible. The game total domination number, γtg (G), of G is the minimum possible number of vertices chosen when Dominator starts the game and both players play according to the rules. The Staller (G), of G is the number of vertices chosen when Staller start game total domination number, γtg starts the game and both players play optimally. We refer to the Dominator-start total domination game and the Staller-start total domination game simply as the Dominator-start game and Staller-start game, respectively. We denote the sequence of moves played in the Dominator-start game by d1 , s1 , d2 , s2 , . . ., where di and si are the ith moves played by Dominator and Staller, respectively, where i ≥ 1. Further, we denote the sequence of moves played in the Staller-start game by s1 , d1 , s2 , d2 , . . ., where si and di are the ith moves played by Staller and Dominator, respectively, where i ≥ 1. Adopting the terminology in [13], a partially-dominated graph is a graph in which we suppose that some vertices have already been totally dominated, and need not be totally dominated again to complete the game. Given a graph G and a subset S of vertices of G, we denote by G|S the partially-dominated graph in which the vertices of S in G are already totally dominated. We use γtg (G|S) (resp. γtg (G|S)) to denote the number of turns remaining in the game on G|S under optimal play when Dominator (resp. Staller) has the next turn. In [10], the authors present a key lemma, named the Total Continuation Principle, which in particular implies that the number of moves in the Dominator-start game and the Staller-start differ by at most 1; that is, |γtg (G) − γtg (G)| ≤ 1 for every graph G (with no isolated vertex). Lemma 1 ([10]) (Total Continuation Principle) Let G be a graph, and let A, B ⊆ V (G). If B ⊆ A, (G|A) ≤ γtg (G|B). then γtg (G|A) ≤ γtg (G|B) and γtg (G) is a challenging problem, and is currently only Determining the exact value of γtg (G) and γtg known for paths and cycles [9]. A bound on the game total domination number for general graphs is established in [11] where it is shown that if G is a graph on n vertices in which every component (G) ≤ (4n + 2)/5. This upper bound is contains at least three vertices, then γtg (G) ≤ 54 n and γtg 3 improved in [12] to γtg (G) ≤ 4 n over the class of graphs with the degree sum of adjacent vertices at least 4 and with no two vertices of degree 1 at distance 4 apart in G. Further, the upper bound 8 n over the class of graphs with minimum degree at least 2. is improved in [6] to γtg (G) ≤ 11 In this paper, we study the algorithmic complexity of determining the game total domination number of a given graph. The algorithmic approach to this problem is not at all likely to be polynomial in time. Even worse, it is questionable whether it is NP-hard. It was proven in [2] that the complexity of verifying whether the game domination number of a graph is bounded by a given integer is in the class of PSPACE-complete problems. Hence the decision version of the game domination problem is computationally harder than any NP-complete problem, unless NP=PSPACE. In this paper we use the same known PSPACE-complete problem (notably the POS-CNF problem; see [15] for the complexity result on this problem) to verify that the complexity of the decision version of the total game domination number is PSPACE-complete as well. Formally, we are dealing with the following Game Total Domination Problem:
Game Total Domination Problem Input: A graph G, and an integer m. Question: Is γtg (G) ≤ m? 2
We establish the following result, a proof of which is given in the next section. Theorem 2 Game Total Domination Problem is PSPACE-complete. More precisely, since the reduction from the POS-CNF problem to the game total domination problem can be computed with a working space which is logarithmic in size of a formula in the POS-CNF problem, and since the POS-CNF is log-complete in PSPACE, we derive that Game Total Domination Problem is also log-complete in PSPACE.
2
A Proof of Theorem 2
We present the reduction to the Game Total Domination Problem from the POS-CNF problem, which is known to be log-complete in PSPACE [15]. In POS-CNF we are given a set of variables, and a formula that is a conjunction of disjunctive clauses, in which there is no negations of variables. Two players alternate turns, the first player setting a previously unset variable TRUE, and the second player setting such a variable FALSE. After all variables are set, the first player wins if the formula is TRUE, otherwise the second player wins. In what follows, we use the standard notation [k] = {1, . . . , k}. Let T be the gadget graph shown in Figure 1, where V (T ) = {a, b, b , z, z , x, y, w}. b
x z
y w z
a
b
Figure 1: The gadget, T , used in the construction, representing a variable The following observation summarizes the properties of the graph T needed in the proof of our main result. Note that the rules of the game do not allow any of the players passing a move. However, the graph GF that appears in our main construction consist of several copies of the gadget graph T , and it may happen during the game played on GF that after some moves were played in a subgraph Ti isomorphic to the gadget graph T that one of the players chooses to make a move outside Ti before all vertices of this subgraph are totally dominated. In this case, the other player could be the first one to play again in the subgraph Ti . From the point of view of the graph T , this corresponds to one player passing a move in T . Observation 1 Let T be the gadget graph shown in Figure 1. The following properties hold in T . (a) γtg (T ) = 3, γtg (T |a) = 3, γtg (T ) = 3, γtg (T |a) = 3.
(b) If Dominator starts the game on T with d1 = a, then Staller can ensure three moves are played in T by playing s1 = b. In this case, Dominator can finish the game on T in three steps by playing d2 = x; if, however, Dominator passes his second move, then Staller can ensure four moves are played in T by playing the vertex b .
3
(c) If Dominator starts the game on T with d1 = a, and Staller passes her first move, then Dominator can finish the game on T in only two steps by playing d2 = x (or d2 = w). (d) If Staller starts the game on T with s1 = b, then Dominator can ensure three moves are played in T by playing d1 = x. (e) If Staller starts the game on T with s1 = b, and Dominator responds by playing d1 = a, then Staller can enforce four moves on T by playing s2 = b . (f) If Staller starts the game on T with s1 = b, and Dominator passes his first move, then Staller can enforce four moves on T by playing s2 = b . (g) If Staller starts the game on T with s1 ∈ / {b, b }, then Dominator can finish the game on T in his next move, and so only two moves are played on T . Given a formula F using k variables and n disjunctive clauses, we build a graph GF , having 8(k + 1) + n + 9 vertices, as follows. For each variable Xi (where i ∈ [k]) we add to the graph a copy, Ti , of the gadget graph, T , shown in Figure 1. For each clause Cj (where j ∈ [n]) we add a vertex cj to the graph, and make cj adjacent to a vertex ai , whenever the variable Xi appears in the clause Cj . In addition, we add to the graph a special gadget, T0 . For each vertex u ∈ V (T ) = {a, b, b , x, y, z, z , w}, we denote the associated vertex in the ith gadget Ti by ui , where i ∈ {0} ∪ [k]. We make the vertex a0 in the gadget T0 (corresponding to the vertex a in the gadget T ) adjacent to all vertices cj , for j ∈ [n]. Finally, we add a vertex v and make it adjacent to all vertices ai , where i ∈ {0} ∪ [k], and add vertices e1 , . . . , e4 , f1 , . . . , f4 , and make each vertex ei adjacent to both vertices v and fi for all i ∈ [4]. We note that the set {c1 , . . . , cn } is an independent set in GF . To illustrate the construction of the graph GF , consider the formula F = C1 ∧ C2 ∧ C3 , where C1 = X1 ∨ X3 , C2 = X2 ∨ X4 , and C3 = X3 ∨ X4 . In this case, there are k = 4 variables and n = 3 disjunctive clauses. The associated graph GF is illustrated in Figure 2, where in each gadget, Ti , where i ∈ {0} ∪ [4], we only label the vertex ai (corresponding to the vertex a in the gadget T ). f1 e1
f2 e2
f3 e3
f4 e4
v
X1
a0
a1
X2
X3
X4
a3
a2
c1
c2
a4
c3
Figure 2: The graph, GF , associated with the formula F = (X1 ∨ X3 ) ∧ (X2 ∨ X4 ) ∧ (X3 ∨ X4 ). 4
The following result implies the proof of our main theorem. We say that a gadget is opened if a move has been played in that gadget. Theorem 3 Let F be a formula with k variables, which is a conjunction of n disjunctive clauses, and let GF be the corresponding graph. Player 1 has a winning strategy for F in the POS-CNF game if and only if γtg (GF ) ≤ 3k + 8. Proof. We first assume that Player 1 has a winning strategy for F in the POS-CNF game, and describe a strategy for Dominator that guarantees that at most 3k + 8 moves will be played in the total domination game played in the graph GF . Dominator plays as his first move, d1 , the vertex v, thus totally dominating all vertices ai , where i ∈ {0} ∪ [k] and all vertices ei , where i ∈ [4]. This implies that no vertex cj , where j ∈ [n], and also no vertex fj , where j ∈ [4], is a legal move in the remaining part of the game. (Eventually, each vertex ei will have to be played in the game to totally dominate the vertex fi for all i ∈ [4]. If during the game, Staller plays the vertex ei , then on the next move of Dominator he plays a vertex ej that has not yet been played for some j = i. By this strategy, the vertices ei have no effect on the rest of the game, and we may assume that they are played at the end of the game.) We remark that whenever Staller makes a move which is the first move played from a gadget, Ti , for some i ∈ {0} ∪ [k], then, by Observation 1(g), it is in her best interest to play the vertex bi or the vertex bi . Renaming vertices, if necessary, we may assume that Staller plays the vertex bi in such circumstances. Suppose that Staller plays as her first move, s1 , a vertex from the gadget Ti , for some i ∈ {0} ∪ [k]. By our earlier assumption, s1 = bi . Dominator responds by playing as his second move, d2 , the vertex xi from the same gadget Ti . This strategy of Dominator implies that all vertices in the gadget Ti are totally dominated by the played vertices, except for exactly one vertex, namely the vertex wi . Thus, only one additional vertex can be played among the vertices in Ti , and such a vertex is necessarily a neighbor of wi (in order for the move to be legal). If Staller plays as her second move, s2 , a vertex from the gadget Ti , for some i ∈ {0} ∪ [k] \ {i}, then, by our earlier assumption, s2 = bi and Dominator responds by playing as his third move, d3 , the vertex xi from the same gadget Ti . Dominator continues with this strategy by always responding to a move bj of Staller which is the first move played in some gadget Tj , where j ≥ 1, by playing the vertex xj in the same gadget Tj . Renaming variables and clauses, if necessary, we may assume for notational convenience, that Staller’s first moves s1 , . . . , s are the vertices b1 , . . . , b , from the first distinct gadgets T1 , . . . , T , and Dominator’s first + 1 moves d1 , d2 , . . . , d+1 are the vertices v, x1 , . . . , x . Further, we assume that is the largest such integer. Possibly, = k. In view of the preceding arguments, we may therefore assume, renaming variables and clauses if necessary, that for ≥ 0, Staller’s first + 1 moves s1 , . . . , s+1 are the vertices b0 , . . . , b , from the first + 1 gadgets T0 , T1 , . . . , T , and Dominator’s first + 2 moves d1 , d2 , . . . , d+2 are the vertices v, x0 , . . . , x . Further, we may assume that is the largest such integer. Possibly, = 0 or = k. By our choice of the integer , Staller’s ( + 2)nd move is played in one of the previously opened gadgets Ti , where i = 0 or i ∈ [] and ≥ 1. As observed earlier, this move of Staller is a neighbor of wi , and all vertices of Ti are now totally dominated by the played vertices. If i = 0 (and so, ≥ 1 and i ∈ []), then Dominator responds to this move of Staller by playing as his ( + 3)rd move the vertex a0 . As before, by Observation 1(a), Dominator can make sure the game takes at most 3k + 8 moves to complete. Hence, we may assume that Staller’s ( + 2)nd move is played in the gadget T0 and is not the vertex a0 . Thus, three vertices are currently played from T0 and all vertices of T0 are totally dominated. We remark that at this stage of the game, none of the clause vertices, cj , is yet totally dominated. Dominator now plays as his ( + 3)rd move the vertex at in the gadget Tt , where Xt is the first variable Player 1 would set TRUE in the POS-CNF game. If t ∈ [], then all vertices of Tt are now totally dominated by the played vertices. If t ∈ / [], then < k and t ≥ + 1. In this case, if Staller 5
responds by playing her ( + 3)rd move in the gadget Tt , then Dominator plays the vertex xt in reply to Staller’s move, making all vertices of Tt totally dominated, and thereby ensures the game takes at most 3k + 8 moves to complete in view of the preceding arguments. Hence, Staller plays as her ( + 3)rd move either the first move played in some gadget Ts that has not previously been opened, or a third move in a previously opened gadget Tr . In the former case, Dominator responds to Staller’s move by playing the vertex xs . In this case, only the vertex ws remains totally undominated in Ts ; hence, by the Total Continuation Principle, if Dominator is going to be the last to play in Ts , he would choose as , while Staller would in such event choose some other neighbor of ws . In the latter case, that is, if Staller plays the third move in a previously opened gadget Tr , such a move of Staller is a neighbor of wr , and all vertices of Tr are now totally dominated by the played vertices. Dominator considers such a move of Staller as Player 2 setting the associated variable Xr FALSE in the POS-CNF game, and follows the POS-CNF winning strategy of Player 1. Thus, if Xp is the next variable that Player 1 would set TRUE in the POS-CNF game in response to the previous move by Player 2 which set the variable Xr FALSE, then in the total domination game played in GF , Dominator would play as his next move the vertex ap . With this strategy of Dominator, he can guarantee that at most 3k + 8 moves are played in order to complete the game. Now we assume that Player 2 has a winning strategy for F in the POS-CNF game, and describe a strategy for Staller that guarantees that at least 3k + 9 moves will be played in the total domination game played in the graph GF . The base part of Staller’s strategy is that whenever Dominator plays in some gadget Ti , her response is to reply with a move in the same gadget Ti , unless Dominator played the last (in most cases the third) move in Ti , which makes all vertices of Ti totally dominated. By Observation 1(a) and the structure of GF , this implies that at least three moves will be played in each Ti . If during the game, Dominator plays the vertex ei , then on the next move of Staller she plays a vertex ej that has not yet been played for some j = i. By this strategy, the vertices ei have no effect on the rest of the game, and we may assume as in the first part of the proof that they are played at the end of the game. First let us assume that as his first move, d1 , Dominator chooses the vertex v, and so d1 = v. Staller replies in T0 by playing the move s1 = b0 . If Dominator as the next move or at some later point in the game chooses a0 as his move, then, by Observation 1(e), Staller can enforce four moves are played in T0 by choosing b0 as her next move; this, together with her basic strategy, by which (at least) three moves will be played in each Ti , already guarantees that at least 3k + 9 moves will be played in the game (counting also four moves played on the vertices ei for i ∈ [4]). If Dominator plays some vertex in T0 different from a0 as his second move, then Staller responds by playing a legal move in T0 different from a0 . In this way, all vertices of T0 are totally dominated, and three vertices of T0 are played. Further, the vertex a0 is not one of the played vertices in T0 . Thus, we may assume that the second move of Dominator is d2 = x0 and Staller’s second move is s2 = y0 . The third move of Dominator is in Tt for some t ∈ [k], hence, noting that in the graph GF the set N (wt ) is contained in N (xt ), while N (zt ), N (zt ) and N (yt ) are all contained in N (at ), by the Total Continuation Principle we may assume that Dominator chooses one of the vertices from {at , bt , bt , xt }. By her strategy, Staller follows by choosing a vertex from Tt as her third move. If d3 = at , then Staller can ensure that after her third move the vertex wt is not totally dominated. If, on the other hand, d3 = at , Staller responds by playing s3 = bt . Following the first alternative in the previous paragraph, Dominator may open several other gadget subgraphs Ti by playing a vertex different from ai , whence Staller follows by playing another move in Ti that ensures that wi is not totally dominated after her move. Continuing in this way, suppose that + 1 gadget subgraphs are opened for some ≥ 0. Renaming variables and clauses if necessary, we may assume that T0 , T1 , . . . , T are the opened gadgets, where all vertices of T0 are totally dominated, while in Ti , for i ∈ [], the vertex wi is not yet totally dominated. Further we 6
assume that is the largest such integer, and possibly, = k. Hence, the ( + 3)rd move of Dominator is the vertex at , for some t ∈ [k]. Possibly, t ≤ in which case all vertices of the gadget subgraph Tt are totally dominated. We proceed further with the following claim. Claim A During the game, if Dominator opens two different gadgets Ti by playing vertex ai without first ensuring that all vertices in one of these two gadgets are totally dominated, then Staller guarantees that at least 3k + 9 moves will be played. Proof. Suppose that as and at are moves of Dominator and these moves open the gadgets Ts and Tt without first ensuring that all vertices in one of these two gadgets are totally dominated. The corresponding responses of Staller are bs and bt , by her basic strategy. Further assume that xs and xt have not yet been played, and not all vertices of Ts and Tt are totally dominated. Then, at some point, Dominator will have to choose a vertex from a previously opened gadget, thereby totally dominating every vertex from that gadget. Renaming indices if necessary, we may assume that among the two gadgets, Ts and Tt , Dominator first returns to Tt by playing the vertex xt as the third move played in Tt . Now, Staller responds by playing the vertex bs in the gadget Ts , which, by Observation 1(f), implies that four moves are played in Ts ; hence at least 3k + 9 moves will be played in the game, as desired. This completes the proof of the Claim A. (2) We remark that in view of our preceding arguments, it is only in Staller’s interest if Dominator opens gadgets Ti by playing a vertex different from ai . More precisely, if Dominator opens two different gadgets Ti by playing wherever he wishes in these gadgets and without first ensuring that all vertices in one of these two gadgets are totally dominated, then Staller guarantees that at least 3 moves will be played in each gadget (and if in both gadgets vertices ai were played, then Staller can enforce that at least 3k + 9 moves will be played in total). We may therefore assume that whenever Dominator opens a gadget Ti by playing ai , for some i ∈ [k], and Staller, according to her strategy, responds by playing the vertex bi as her next move, then Dominator immediately responds to Staller’s move by playing the vertex xi that belongs to the gadget Ti , thereby totally dominating every vertex from Ti . We now return to the proof of Theorem 3. Recall that the ( + 3)rd move of Dominator is the vertex at , for some t ∈ [k]. Staller considers the move at of Dominator as if Player 1 would set variable Xt TRUE. We consider the case when t ≤ and the case when t > separately. Case 1. Suppose t ≤ . In this case, all vertices of Tt are totally dominated after the move d+3 = at . Staller responds by playing in Tr , where the optimal response of Player 2 in the POSCNF game to Player 1 setting the variable Xt TRUE, is setting the variable Xr FALSE. There are two possibilities for Staller’s move. If r ≤ , then Staller plays a vertex in Tr − ar , which totally dominates wr , and all vertices of Tr are then totally dominated. On the other hand, if r > , Staller plays br . By Observation 1(e), in order to avoid four moves being played in the gadget subgraph Tr , Dominator responds to Staller’s move, br , by playing the vertex xr . Staller in turn responds to Dominator’s move by playing the vertex yr , thereby totally dominating every vertex from the gadget Tr . Case 2. Suppose t > . In this case, Dominator’s ( + 3)rd move opens the gadget Tt . Staller responds by choosing as her ( + 3)rd move the vertex bt . By our assumption following the proof of Claim A, Dominator responds in Tt by playing xt . Thus, Dominator’s ( + 4)th move plays the vertex xt , thereby totally dominating all vertices of Ti . The response of Staller, s+4 , is analogous as in Case 1. Notably, Staller responds by playing in Tr , where the optimal response of Player 2 in the POS-CNF game to Player 1 setting the variable Xt TRUE, is setting the variable Xr FALSE. As in Case 1, there are two possibilities with respect to r being bigger than or not, and they are dealt with in the same way as in Case 1. 7
In the remainder of the game, Dominator may choose a vertex at in some Tt in which case Staller responds in the same way as in one of the above cases, depending on whether Dominator opens the gadget Tt by playing at or whether Tt is already opened prior to his move. In both cases, by the same reasoning as above, Staller responds in such a way that one of the gadget subgraphs Tr becomes totally dominated without ar being played. On the other hand, Dominator may also choose to open a new gadget subgraph Tt without playing at . In this case, Staller only responds in that gadget subgraph, and it is still not determined, whether at will be played or not. (Such moves are not reflected in the POS-CNF game, which is simultaneously considered by Staller.) By Staller’s strategy, the game in GF reflects the POS-CNF game in F. We derive that after all 3(k + 1) moves are played in the gadgets Ti , i ∈ {0} ∪ [k], there is still a clause vertex cj , which is not yet totally dominated. Hence, at least one additional vertex must be played in order to make sure that the clause vertices are all totally dominated. Thus, including the first move d1 = v of Dominator, and moves on vertices e1 , e2 , e3 , e4 , at least 3k + 9 moves are played in the game. Finally, let us assume that the first move, d1 , of Dominator is not the vertex v. By the Total Continuation Principle, it is never in Dominator’s best interests to play a clause vertex cj , since in this case he can always do at least as well by playing the vertex v which totally dominates a superset of vertices in N (cj ). As discussed earlier, we may assume that the vertices ei are played at the end of the game. Indeed, if Dominator plays the vertex ei , then Staller can only benefit from this move and continue to follow the base part of her strategy as described earlier. Hence we may assume that Dominator starts to play in one of the gadgets Ti , for some i ∈ {0} ∪ [k]. Staller now plays as her first move, s1 , the vertex f1 . There are several cases with respect to the next few moves of Dominator. He could continue to play in gadget subgraphs, in which case, Staller would choose f2 , f3 and eventually also f4 . Alternatively, Dominator could in one of his first four moves choose v and prevent fi as a legal move. Let us only consider the first case where Staller plays the vertices f1 , . . . , f4 on her first four moves. We remark that the other cases are similar, only slightly shorter in argumentation. By Observation 1(c), Dominator can finish the game in the gadget subgraph Ti in only two moves, if Staller chooses not to respond to Dominator’s move in Ti . Hence, after the first five moves of Dominator the following can occur: • Five gadgets Ti are opened with only one move in each of the gadgets played by Dominator; • Four gadgets Ti are opened, where all vertices in one gadget are totally dominated, whereas in other three gadgets Dominator played only one move; • Three gadgets Ti are opened, where two gadgets have all vertices totally dominated, whereas in one gadget only one move of Dominator was played. In addition, all vertices f1 , . . . , f4 were chosen by Staller, and it is Staller’s turn to play. In her next move, s5 , Staller plays in one of the opened but not yet totally dominated gadgets, thereby ensuring at least three moves will be played in this gadget (using Observation 1(a)). Regardless of which of the three possibilities above occur as a consequence of the first five moves played by Dominator, Staller can guarantee with her strategy that in at most two gadgets are exactly two moves played, while in the remaining k − 1 gadgets at least three moves are played. In addition, all eight vertices e1 , . . . , e4 , f1 , . . . , f4 are played, implying that at least 2 × 2 + 3(k − 1) + 8 = 3k + 9 moves will be played in the game, which makes the final case of Staller’s strategy complete.
3
Concluding Remarks
In this paper we proved the PSPACE-completeness of Game Total Domination Problem by using a reduction from the POS-CNF problem, in which a formula F is given, to the total domination 8
game of a special graph GF . It is easy to see that the reduction we used can be computed with a working space of logarithmic size with respect to the entry, making this problem log-complete in PSPACE. An analogous result, i.e., PSPACE-completeness, is true also for the Staller-start version of the game total domination number. Moreover, even the same construction can be used, translating a formula F to the graph GF , and it can be proved, that Player 1 has a winning strategy for F in (GF ) ≤ 3k + 9. Let us give some brief ideas about the proof the POS-CNF game if and only if γtg of this result. It is easy to see that when Player 1 has a winning strategy for F, then Dominator can ensure at most 3k + 9 moves are played in GF , when Staller starts the game. Indeed, after any first move of Staller, the graph GF becomes partially totally dominated, and by combining the Total Continuation Principle with the proof of Theorem 3, he can ensure that only 3k + 8 moves are played after the first move of Staller. On the other hand, when Player 2 has a winning strategy for a formula F in POS-CNF, Staller’s strategy is to start in a clause vertex cj (note that there exists a clause vertex that is not adjacent to all vertices ai , hence v is still a legal move after Staller’s first move). If Dominator’s response is to play v, then the game already reduces to the previous version of the game in GF , where Staller can ensure 3k + 9 moves are played, altogether 3k + 10 moves counting her first move. However, if Dominator responds in some gadget, then the situation becomes similar to the final part of Staller’s strategy in the proof above, where Staller chooses vertices fi .
Acknowledgements We are grateful to the reviewers for their helpful remarks and suggestions. B. Breˇsar is supported in part by the Slovenian Research Agency (ARSS) under the grants P1-0297, J1-7110, and N1-0043. Research of M. A. Henning is supported in part by the South African National Research Foundation and the University of Johannesburg.
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3 4 -Conjecture.
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