solitaire clobber on circulant graphs
TRANSCRIPT
Discrete Mathematics 329 (2014) 33–41
Contents lists available at ScienceDirect
Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Solitaire clobber on circulant graphs✩
Telma Pará a,∗, Simone Dantas b, Sylvain Gravier ca COPPE, Federal University of Rio de Janeiro, Brazilb IME, Fluminense Federal University, Brazilc CNRS/UJF, Institut Fourier, SFR Maths à Modeler, France
a r t i c l e i n f o
Article history:Received 1 May 2012Received in revised form 31 March 2014Accepted 3 April 2014Available online 26 April 2014
Keywords:Graph theoryCombinatorial gamesSolitaire clobber
a b s t r a c t
Solitaire Clobber is a one-player combinatorial game on graphs. Each vertex of a graph Gstarts with a black or a white stone. A stone on one vertex can clobber an adjacent stone ofthe opposite color, removing it and taking its place. The goal is to minimize the number ofstones remaining when no further move is possible. An initial configuration is k-reducibleif it can be reduced to k stones. A graph is strongly 1-reducible if, for any vertex v, any initialconfiguration that is not monochromatic outside v can be reduced to one stone, on v, ofeither color. Every such graph has a Hamiltonian path ending at v. For the path Pn, we provethat the rth distance power P r
n is strongly 1-reducible when r ≥ 3 but not when r = 2 (P2n
is 2-reducible). As a consequence, circulant graphs containing edges of lengths 1, 2, and 3are strongly 1-reducible; we show also that those containing C2
n are 1-reducible.© 2014 Elsevier B.V. All rights reserved.
1. Introduction
Clobber is a combinatorial game that was first introduced by Albert et al. [1] at the seminar on Algorithmic andCombinatorial Game Theory in Dagstuhl. Solitaire Clobber (SC) was introduced by Demaine et al. [4] as a one-player gamewhere the player is not forced to alternate black and white stone moves. The rules of Solitaire Clobber are: each vertex ofa graph G starts with a black or a white stone. A stone on one vertex can clobber an adjacent stone of the opposite color,removing it and taking its place. The goal is to minimize the number of stones remaining when no further move is possible.
We consider simple graphs G with vertex set V (G) and edge set E(G). Usually, we set n = |V (G)|. A configuration Φ
on G is a mapping Φ : V (G) → { t, ❞}. A configuration having stones of only one color is monochromatic. Otherwise, itis non-monochromatic. We say that a configuration Φ on a graph G is k-reducible (for a positive integer k) if there exists asuccession of moves that leaves at most k stones on G. The reducibility value rv(G, Φ) is the smallest integer k for which(G, Φ) is k-reducible.
It was proved in [6] that solitaire clobber is NP-hard by reducing the NP-complete Hamiltonian path to solitaireclobber.
solitaire clobberInstance: (G, Φ), where Φ is a configuration on a graph G.Question: (G, Φ) is 1-reducible?In order to reduce the Hamiltonian path to solitaire clobber, we consider (G, Φ) such that Φ(v) = tfor some vertex
v ∈ V (G) and Φ(u) = ❞for all u ∈ V (G) − v. The graph G with this configuration is clearly 1-reducible if and only if there
✩ This work was partially supported by CAPES/COFECUB, CNPq and FAPERJ.∗ Corresponding author.
E-mail addresses: [email protected] (T. Pará), [email protected] (S. Dantas), [email protected] (S. Gravier).
http://dx.doi.org/10.1016/j.disc.2014.04.0060012-365X/© 2014 Elsevier B.V. All rights reserved.
34 T. Pará et al. / Discrete Mathematics 329 (2014) 33–41
exists a Hamiltonian path from v to some other vertex. Since the Hamiltonian path is NP-complete on graphs in generaland, in particular, on grid graphs [7], solitaire clobber is NP-hard on such graphs. We note that a grid graph is a finite graphembedded in the Euclidean plane such that the vertices have integer coordinates and two vertices are connected by an edgeif and only if their Euclidean distance is equal to one.
Solitaire Clobber was investigated by Blondel et al. [3] on paths and cycles and by Beaudou et al. [2] on trees. If G is a path(resp. cycle) with n vertices, then the reducibility value is at most
n2
(resp.
n3
), and these bounds are sharp, i.e., there are
configurations that attain these bounds. For paths and cycles, the reducibility value can be computed in linear time. If G is atree, then it can be computed in time O(n3) for any configuration. Dorbec et al. [5] proved for Hamming graphs (Cartesianproducts of any number of complete graphs) that every non-monochromatic configuration is 1-reducible, except that forhypercubes the reducibility value may be 2.
In 2008, Dorbec et al. [5] introduced a more complex question about Solitaire Clobber: is it true that for every vertex vof G, for every configuration on G (provided G − v is non-monochromatic), for every color c (black or white), there exists away to play that yields a single stone of color c on v? If the answer is yes, then the graph G is strongly 1-reducible. We notethat the class of strongly 1-reducible graphs is contained in the class of graphs for which there exists a Hamiltonian pathwith ending point at v, for all v ∈ V (G). More precisely, a configuration that is monochromatic except for one vertex v ofthe opposite color is 1-reducible if and only if there is a Hamiltonian path starting at v.
Strong reducibility was studied for certain classes of graphs: all Hamming graphs are strongly 1-reducible excepthypercubes and the Cartesian product of K2 with K3, and all complete graphs with at least three vertices are strongly 1-reducible. Also, if G is a strongly 1-reducible graph having at least four vertices, then the Cartesian product of G with anycomplete graph is strongly 1-reducible [5].
We know that the n-vertex cycle Cn is not strongly 1-reducible and the complete graph Kn is strongly 1-reducible forn > 2. How does strong reducibility behave between these two families? To begin this study, we focus our attention on theclass of circulant graphs, or more specifically powers of cycles.
Before stating our results, we need some definitions. The distance between vertices u and v, written dG(u, v), is thenumber of edges in a shortest path from u to v. The rth power of a graph G, written Gr , is the graph with vertex set V (G) inwhich u and v are adjacent if and only if dG(u, v) ≤ r . For A ⊆
1, . . . ,
n2
, the circulant graph C(n, A) is the graph whose
vertices are the congruence classes modulo n, with vertices adjacent if and only if the difference between them lies in A. Inparticular, every power of a cycle is a circulant graph.
In this paper, we prove that if r ≥ 3, then P rn is strongly 1-reducible. On the other hand, P2
n is not strongly 1-reducible.As a direct corollary, C(n, A) is strongly 1-reducible when {1, 2, 3} ⊆ A. We prove that P2
n is 2-reducible, and we presentconfigurations on P2
n for which the reducibility value is 2. We prove that C rn is strongly 1-reducible for r ≥ 1, and we give
bounds for the reducibility value of Solitaire Clobber played on certain classes of circulant graphs. For C(n, A), we provethat if {d, 2d} ⊆ A and gcd(n, d) = 1, then rv(C(n, A), Φ) ≤ 1, for any non-monochromatic configuration Φ . Also ifA = {d1, . . . , dk} such that d1, . . . , dk are odd and n is even, then rv(G, Φ) ≤ min
n − 2|A|, ⌈
n3⌉
.
Finally we conclude with the following open problems.
Problem 1. Characterize for which values of d there exists a configuration on C(n, {1, d})with d > 2 that is not 1-reducible.
According to the results of Blondel et al. [3], the reducibility value of C(n, A) when |A| = 1 is not bounded by a constant(independent of n). From our result we note that rv(C(n, {1, d}), Φ) = 1, and we suggest the following problem.
Problem 2. Does there exist some constant k such that rv(C(n, {d1, d2}), Φ) ≤ k, where Φ is any non-monochromaticconfiguration?
2. Strong reducibility
We denote by Φ(v) the color of the stone on v, where c ∈ { t, ❞} . We also say that v is labeled c. Let X ⊆ V (G). Given aconfiguration Φ on G, we write ΦX for the restriction of the configuration Φ to G[X]. A color c is rare on G if only one vertexis labeled c. We say that (G, Φ) is (1, v, c)-reducible if (G, Φ) is 1-reducible to v with color c , that is, if there exists a way toplay that yields only one stone of color c , located on v. Therefore G is strongly 1-reducible if (G, Φ) is (1, v, c)-reducible forall c and for all v such that Φ is non-monochromatic on G − v.
For convenience, we may say that vertex v clobbers vertex u instead of talking about the corresponding stones.We begin by proving our results concerning the strong reducibility of powers of paths. We refer to Figs. 1–3, where ‘‘x’’
marks vertices whose stones are already clobbered.In the following, we will consider graphs of order at least 3.
Theorem 3. For 3 ≤ r ≤ n − 1, the graph P rn is strongly 1-reducible.
Proof. Let A and B be two disjoint subsets of consecutive vertices of V (P rn), called segments, and vi be a vertex of V (P r
n)between A and B such that V (P r
n) = A ∪ {vi} ∪ B. We prove, by induction on n, that (G, Φ) is (1, vi, c)-reducible for everycolor c , every vertex vi of G, and every configuration on P r
n such that Φ is non-monochromatic on P rn − vi. We observe that
T. Pará et al. / Discrete Mathematics 329 (2014) 33–41 35
(a) B non-monochromatic. (b) B monochromatic.
Fig. 1. Examples for P rn, r ≥ 3, with A non-monochromatic.
Fig. 2. Example for P rn, r ≥ 3, with monochromatic segments A and B and c = Φ(v).
Fig. 3. Example for P rn, r ≥ 3, with monochromatic segments A and B and c = Φ(v).
the theorem is valid for n = 4, since by [5], all complete graphs are strongly 1-reducible, and it is easy to verify for n = 5.Now we consider three cases.
Case (i): Let ΦA and ΦB be two non-monochromatic configurations (see Fig. 1(a)). First, we play on G[A ∪ {vi}]. Since ΦAis non-monochromatic, by the induction hypothesis, (G[A ∪ {vi}], ΦA∪{vi}) is (1, vi, c)-reducible, for all c . Now we play onG[B∪ {vi}]. Since ΦB is non-monochromatic, again, by the induction hypothesis, (G[B∪ {vi}], ΦB∪{vi}) is (1, vi, c)-reducible,for all c.
Case (ii): Let ΦA be non-monochromatic, and let ΦB be monochromatic. Without loss of generality, suppose that everyvertex of B is white. See an example in Fig. 1(b). First we play on G[A]. IfΦ(A−vi−1) is non-monochromatic, by the inductionhypothesis, (G[A], ΦA) is (1, vi−1, t)-reducible. Vertex vi−1 clobbers vi+1. Now, Φ ′
B is non-monochromatic. By the inductionhypothesis, (G[B∪ {vi}], ΦB∪{vi}) is (1, vi, c)-reducible. Case Φ(A− vi−1) is monochromatic is similar by considering vertexvi−2.
Case (iii): LetΦA andΦB bemonochromatic.We have two subcases: segments A and B have the same color, and segmentsA or B have opposite colors. The first subcase will not be considered because Φ(G − vi) would be monochromatic. In thesecond subcase, without loss of generality, we may assume that segment A is black, segment B is white and vi is black (seeFig. 2). Let A′
= A− vi−1 and B′= B− vi+1. First assume that c = Φ(vi). We proceed as follows: vi−1 clobbers vi+2, and vi+1
clobbers vi−2. We note that Φ ′(B′− vi+3) is non-monochromatic. By the induction hypothesis, (G[B′
], ΦB′) is (1, vi+3, ❞)-reducible. We also note that Φ ′(A′
− vi−3) is non-monochromatic. By the induction hypothesis, (G[A′], Φ ′
A′) is (1, vi−3, t)-reducible. The last moves are: vi+3 clobbers vi, and vi−3 clobbers vi. Now assume that c = Φ(vi). First, vi+1 clobbers vi−2. Wenote that Φ ′
A′ is non-monochromatic. By the induction hypothesis, (G[A], Φ ′
A) is (1, vi−1, t)-reducible. Next, vi−1 clobbersvi+2. We observe that Φ ′
B′ − vi+3 is non-monochromatic. By the induction hypothesis, (G[B′], Φ ′
B′) is (1, vi+3, ❞)-reducible.Finally, vi+3 clobbers vi and we conclude the proof. �
The next result is a direct consequence of Theorem 3 and P rn ⊂ C r
n .
Corollary 4. For 3 ≤ r ≤ n − 1, the graph C rn is strongly 1-reducible. �
Next we study the strong reducibility of P rn when r = 2. We note that for configurations Φ of P2
4 with a rare color tandvertices v such that Φ(v) = twe have: if Φ = ❞❞❞t, then (P2
4 , Φ) is (1, v, c)-reducible, for any v and for any c; and ifΦ = ❞❞t❞, then (P2
4 , Φ) is not (1, v, ❞)-reducible when v is the first vertex. In what follows, we present results concerningthe reducibility of P2
n .
36 T. Pará et al. / Discrete Mathematics 329 (2014) 33–41
Fig. 4. Rules of playing on the initial vertices.
(a) Case n = 4. (b) Case n = 5.
Fig. 5. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3) = ◦ ◦ • •.
Fig. 6. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3) = ◦ ◦ • •, n = 6.
We prove that (P2n , Φ) is 2-reducible. First, in Lemma 5, we show that if the first four vertices are non-monochromatic,
then (P2n , Φ) is 1-reducible. Otherwise, in Lemma 8, we show that P2
n is 2-reducible.
Lemma 5. If Φ is a non-monochromatic configuration on P2n , where the first four vertices are non-monochromatic, then (P2
n , Φ)is 1-reducible.
Proof. Weprove by induction on n that under these conditions (P2n , Φ) is 1-reducible. If n ≤ 2, then clearly Lemma 5 is true.
Refer to Fig. 4, where we specify various types of moves that we call rules R1 through R4. The ‘‘x’’ in Fig. 4 marks verticeswhose stones are already clobbered, and the numbers on arrows indicate the order of moves.
First assume Φ(0)Φ(1)Φ(2) = t❞❞. Apply rule R2. This leads to a configuration Φ ′ on P2n−1 satisfying Φ ′(0) = Φ ′(1),
and we conclude by applying the induction hypothesis. Now assume that Φ(0)Φ(1)Φ(2) = ❞t❞. If Φ(3) = t, then applyR3( ❞). Otherwise, if Φ(3) = ❞, then apply R3( t). In both cases, we conclude by applying the induction hypothesis.
Second, we consider Φ(0)Φ(1)Φ(2) = ❞❞t. Let n ≥ 7. Suppose that Φ(3) = t. If Φ(4)Φ(5)Φ(6) = ❞❞❞, then applyR1( ❞). Move from vertex 1 to 3 and apply the induction hypothesis. If Φ(4)Φ(5)Φ(6) = ❞❞❞, then apply R1( t). Move fromvertex 4 to 3. Move from vertex 1 to 3, then from vertex 3 to 5, and then conclude by applying the induction hypothesis. Thesolution of the remaining small cases of n is presented in Figs. 5 and 6.
Now suppose Φ(3) = ❞. If Φ(4)Φ(5)Φ(6) = ttt, then apply R1( t). Move from vertex 1 to 3 and apply the inductionhypothesis. If Φ(4)Φ(5)Φ(6) = ttt, then apply R1( ❞). Move from vertex 4 to 3. Move from vertex 1 to 3, next from 3 to 5and again conclude by applying the induction hypothesis. The remaining small cases of n are presented in Figs. 7 and 8.
Finally, assume Φ(0)Φ(1)Φ(2) = ❞❞❞. By hypothesis, we have Φ(3) = t. Let n ≥ 8. Suppose that Φ(4) = ❞. IfΦ(5)Φ(6)Φ(7) = ttt, then apply R4( t). Move from vertex 2 to 4 and apply the induction hypothesis. IfΦ(5)Φ(6)Φ(7) =ttt, then apply R4( ❞). Move from vertex 5 to 4. Move from vertex 2 to 4 and from 4 to 6, and conclude by applying theinduction hypothesis. The remaining cases are presented in Figs. 9 and 10.
Now, suppose Φ(4) = t. If Φ(5)Φ(6)Φ(7) = ❞❞❞, then apply R4( ❞). Move from vertex 2 to 4 and apply the inductionhypothesis. If Φ(5)Φ(6)Φ(7) = ❞❞❞, then apply R4( t). Move from vertex 5 to 4. Move from vertex 2 to 4 and from 4 to 6,and then apply the induction hypothesis. We finish the proof by checking the small cases depicted in Figs. 11 and 12. �
As a consequence of Lemma 5 (Corollary 6), all powers of cycles C rn, r ≥ 2, are 1-reducible.
Corollary 6. If Φ is a non-monochromatic configuration on C(n, A), {1, 2} ⊆ A, then (C(n, A), Φ) is 1-reducible.
T. Pará et al. / Discrete Mathematics 329 (2014) 33–41 37
(a) Case n = 4. (b) Case n = 5.
Fig. 7. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3) = ◦ ◦ • ◦.
Fig. 8. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3) = ◦ ◦ • ◦, n = 6.
(a) Case n = 5. (b) Case n = 6.
Fig. 9. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3)Φ(4) = ◦ ◦ ◦ • ◦.
Fig. 10. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3)Φ(4) = ◦ ◦ ◦ • ◦, n = 7.
(a) Case n = 5. (b) Case n = 6.
Fig. 11. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3)Φ(4) = ◦ ◦ ◦ • •.
Proof. Since {1, 2} ⊆ A and Φ is a non-monochromatic configuration, it is always possible to find P2n as a subgraph of
C(n, A), with the first four vertices non-monochromatic. By Lemma 5, the proof follows. �
There exist configurations on (P2n , Φ) that are not 1-reducible, for example, the graph P2
10 with configurationttttt❞❞❞❞❞. We leave as an open question the following problem:
Problem 7. Characterize the configurations on P2n that are not 1-reducible.
38 T. Pará et al. / Discrete Mathematics 329 (2014) 33–41
Fig. 12. Remaining small cases with Φ(0)Φ(1)Φ(2)Φ(3)Φ(4) = ◦ ◦ ◦ • •, n = 7.
We show next 2-reducible configurations on P2n .
Lemma 8. If Φ is a non-monochromatic configuration on P2n , where the first four vertices are monochromatic, then (P2
n , Φ)is 2-reducible.
Proof. Let Φ be a non-monochromatic configuration on P2n . Now we consider a configuration Φ on P r
n with four maximalmonochromatic segments B1, B2, B3, and B4 such that |B1| ≥ 4. Without loss of generality, we assume that Φ(v) = t, forv ∈ B1 ∪ B3 and Φ(u) = ❞, for u ∈ B2 ∪ B4. We take 0, 1, . . . , n − 1 to be the vertices in order along the underlying paththat generates P r
n . Refer to Figs. 13–17, where ‘‘x’’ marks vertices whose stones are already clobbered, and the numbers onthe arrows indicate the order of moves.
Claim 9. If |B2| = 1, then (P2n , Φ) is 1-reducible.
Proof of Claim. Let |B1| = 4 and |B2| = 1. If |B3| = 0 (see Fig. 13(a)), thenmove the white stone of B2 in order to clobber allstones of B1 to the left until vertex 0. If |B3| ≥ 2 (see Fig. 13(b)), then move from vertex 4 to 2, from 2 to 0, from 0 to 1, from1 to 3 and from 3 to 5. We reach a configuration Φ ′ where the first four vertices are non-monochromatic, and we concludeby applying Lemma 5 to the remaining stones. Case |B1| > 4 is analogous.
Now consider |B3| = 1. If |B4| = 0 (see Fig. 14(a)), then move from vertex 4 to 5, from 5 to 3, from 3 to 2, from 2 to 1 andfrom 1 to 0. Hence (P2
n , Φ) is 1-reducible. If |B4| = 1 (see Fig. 14(b)), then move from vertex 6 to 5, from 5 to 3, from 3 to 1and from 1 to 2. Move from vertex 0 to 2 and from 2 to 4. Hence (P2
n , Φ) is 1-reducible. If |B4| > 1 (see Fig. 14(c)), thenmovefrom vertex 7 to 5, from 5 to 3, from 3 to 1 and from 1 to 2. Move from vertex 0 to 2, from 2 to 4, from 4 to 6 and from 6 to 8.We reach a configuration Φ ′ where the first four vertices are non-monochromatic, and we conclude by applying Lemma 5to the remaining stones. Case |B1| > 4 is analogous. �
Claim 10. If |B2| = 2 and |B3| = 1, 2, then (P2n , Φ) is 1-reducible.
Proof of Claim. Let |B1| = 4 and |B2| = 2. If |B3| = 0 (see Fig. 15(a)), then move from vertex 3 to 4. Move from vertex 5 to4, from 4 to 2, from 2 to 1 and from 1 to 0. Thus (P2
n , Φ) is 1-reducible. If |B3| > 2 (see Fig. 15(b)), then move from vertex 6to 5. Move from vertex 4 to 2, from 2 to 0, from 0 to 1, from 1 to 3, from 3 to 5 and from 5 to 7. We reach a configuration Φ ′
where the first four vertices are non-monochromatic, and we conclude by applying Lemma 5 to the remaining stones. Case|B1| > 4 is analogous. �
Claim 11. If |B2| = 2 and (|B3| = 1 or |B3| = 2), then (P2n , Φ) is 2-reducible.
Proof of Claim. Let |B1| = 4 and |B2| = 2 (see Fig. 16). Consider the graph P2n . Move from vertex 3 to 4. Move from vertex 5
to 4, from 4 to 2, from 2 to 1 and from 1 to 0. If |B3| = 1 or |B3| = 2 and |B4| = 0, then from vertex 6we reach a configurationΦ ′ where the first four vertices are non-monochromatic, and we conclude by applying Lemma 5 to the remaining stones.If |B4| = 0, then we relabel vertices in the reverse order and get a configuration where the first four vertices are non-monochromatic. Case |B1| > 4 is analogous. �
Claim 12. If |B2| > 2, then (P2n , Φ) is 2-reducible.
Proof of Claim. Consider the graph P2n (see Fig. 17). Move from vertex 4 to 2, from 2 to 1 and from 1 to 0. Move from vertex
3 to 5. Now we have vertex 5 with color black and, since |B2| > 2, vertex 6 is colored white, so from vertex 5 we reacha configuration Φ ′ where the first four vertices are non-monochromatic, and we conclude by applying Lemma 5 to theremaining stones. �
From Lemmas 5 and 8 yield Theorem 13.
Theorem 13. If Φ is a non-monochromatic configuration on P2n , then (P2
n , Φ) is 2-reducible. �
Next, we prove that rv(P2n , Φ) = 2 when Φ is a non-monochromatic configuration on P2
n with only two monochromaticsegments A and Bwith |A|, |B| ≥ 5 such that A∪ B = V (P2
n ), Φ(v) = tfor v ∈ A, and Φ(u) = ❞for u ∈ B. First we need thefollowing results.
T. Pará et al. / Discrete Mathematics 329 (2014) 33–41 39
(a) |B3| = 0. (b) |B3| ≥ 2.
Fig. 13. Case |B2| = 1 for P2n .
(a) |B4| = 0. (b) |B4| = 1. (c) |B4| > 1.
Fig. 14. Case |B2| = 1 and |B3| = 1 for P2n .
(a) |B3| = 0. (b) |B3| > 2.
Fig. 15. Case |B2| = 2 for P2n .
Fig. 16. Case |B2| = 2 and (|B3| = 1 or |B3| = 2) and |B4| = 0 for P2n .
Fig. 17. Case |B2| > 2 for P2n .
Fig. 18. An obstruction.
An obstruction during the play of the game is a configuration in which the subgraph induced by the vertices that stillhave stones has components separated by distance at least 2 in P2
n . This structure is called an obstruction because such aconfiguration prevents reduction to one stone. See Fig. 18, where ‘‘x’’ marks vertices whose stones are already clobberedand Y and Z are segments of vertices. The next result is based on this definition.
Lemma 14. Let Φ be a non-monochromatic configuration on P2n having two monochromatic segments A and B with |A|, |B| ≥ 5
such that A ∪ B = V (P2n ), Φ(v) = tfor v ∈ A, and Φ(u) = ❞for u ∈ B. Every such configuration (P2
n , Φ) is not 1-reducible.
Proof. The proof is based on the fact that all possible moves produce an obstruction.Up to symmetry, we observe that we have three possible startingmoves: J1, J2 or J3 (see Fig. 19). Nowwe show that after
any of these moves, we produce an obstruction.First suppose we play J1 (see Fig. 20). We have the following possible moves: J4, J5, J6 or J7.If we play J4 (see Fig. 21(a)) or J5 (see Fig. 21(b)), then we produce an obstruction.If we play J6 (see Fig. 22), then the only possible moves are J8 or J9. In both cases, we produce an obstruction.
40 T. Pará et al. / Discrete Mathematics 329 (2014) 33–41
Fig. 19. Possible starting moves for P2n .
Fig. 20. Possible moves after playing J1.
(a) Playing J4. (b) Playing J5.
Fig. 21. Playing J4 or J5.
Fig. 22. Playing J6.
Fig. 23. Playing J7.
If we play J7 (see Fig. 23), then the only possible moves are J10 and J11. In both cases, we produce an obstruction.Similarly for moves J2 and J3, in each case we produce an obstruction. Thus (P2
n , Φ) is not 1-reducible. �
Lemmas 8 and 14 yield the following consequence.
Corollary 15. If Φ is a non-monochromatic configuration on P2n having two monochromatic segments A and B with |A|, |B| ≥ 5
such that A ∪ B = V (P2n ), Φ(v) = tfor v ∈ A, and Φ(u) = ❞for u ∈ B, then rv(P2
n , Φ) = 2. �
Lemma 14 obtains configurations on P2n that are not 1-reducible (see Problem 7). Are there other such configurations?
3. Circulant graphs
In this section we first apply our previous results for the class of circulant graphs. We then propose additional results forconfigurations that are not 1-reducible on circulant graphs. We first state consequences of Corollary 6.
Corollary 16. If Φ is a non-monochromatic configuration on the circulant graph C(n, A) with {d, 2d} ⊆ A and gcd(n, d) = 1,then (C(n, A), Φ) is 1-reducible.
Proof. It is well known that C(n, {d, 2d}) ∼= C(n, {1, 2}) when gcd(n, d) = 1. Hence the result follows immediately fromCorollary 6. �
T. Pará et al. / Discrete Mathematics 329 (2014) 33–41 41
Fig. 24. Example for n = 8, d = 3.
Now let G be a bipartite graph with bipartition (S0, S1). Let set S0 (resp. S1) be defined as white (resp. black), and let Φ bea configuration on G. A stone of G is said to be clashing if its color differs from the color of the set to which it belongs. Denoteby δ(G, Φ) the following parameter:
δ(G, Φ) = number of stones + number of clashing stones.
Demaine et al. [4] showed that δ(G, Φ) mod 3 never changes during the game, which proves the following theorem.
Theorem 17 ([4]). If Φ is a configuration such that δ(G, Φ) ≡ 0mod 3, then Φ is not 1-reducible. �
Theorem 17 implies that every 1-reducible configuration of a bipartite graph satisfies δ(G, Φ) ≡ 0 mod 3.We say that an alternated configuration is a configuration on a path in which stones receive colors black and white
alternately. Based on alternated configurations and using δ, we prove the following:
Theorem 18. Let G be a bipartite graph with n vertices. If G has at least one edge, then there is a non-monochromaticconfiguration Φ on G that is not 1-reducible.
Proof. Let (S0, S1) be the bipartition of G and choose uv ∈ E(G) with u ∈ S0. Let Φ ′ be the alternated configuration definedby Φ ′(w) = ❞if and only if w ∈ S0. Let j such that n ≡ j mod 3 with j ∈ {0, 1, 2}. Let Φ ′ be the alternated configuration.
(i) If j = 0, then set Φ = Φ ′. In this case, we have no clashing stone, so δ(G, Φ ′) = n. Hence δ(G, Φ) ≡ 0 mod 3.(ii) If j = 1, then set Φ(x) = Φ ′(x) for all x ∈ V (G) − u, Φ(u) = t. In this case, we have one clashing stone. Thus,
δ(G, Φ) = n + 1 ≡ 0 mod 3. See an example in Fig. 24.(iii) If j = 2, then set Φ(x) = Φ ′(x) for all x ∈ V (G) − uv, and Φ(u) = tand Φ(v) = ❞. Now we have two clashing stones.
Thus, δ(G, Φ) = n + 2 ≡ 0 mod 3.
In these three cases, by Theorem 17, G is not 1-reducible. �
Since C(n, {1, d}) is bipartite when n is even and d is odd, we have the following result:
Corollary 19. For the circulant graph C(n, {1, d}) with n even and d odd, there exist non-monochromatic configurations Φ suchthat (C(n, {1, d}), Φ) is not 1-reducible. �
Acknowledgments
We are grateful to the editor and the anonymous referee for his/her careful reading and valuable contributions, whichhelped to improve this paper.
References
[1] M.H. Albert, J.P. Grossman, R.J. Nowakowsky, D. Wolfe, An introduction to clobber, Integers J. Combin. Number Theory 5 (2) (2005) 1–12. A1.[2] L. Beaudou, E. Duchêne, S. Gravier, A survey about Solitaire Clobber, in: Games of No Chance, Vol. 4, 2014, in press.[3] V.D. Blondel, J.M. Hendrickx, R.M. Jungers, Solitaire Clobber as an optimization problem on words, Integers J. Combin. Number Theory 8 (1) (2008)
1–12. G04.[4] E.D. Demaine, M.L. Demaine, R. Fleischer, Solitaire Clobber, Theoret. Comput. Sci. 313 (2004) 325–338.[5] P. Dorbec, E. Duchêne, S. Gravier, Solitaire Clobber played on hamming graphs, Integers J. Combin. Number Theory 8 (1) (2008) 1–21. G03.[6] E. Duchêne, S. Gravier, J. Moncel, New results about impartial Solitaire Clobber, RAIRO Theor. Inform. Appl. 43 (2009) 463–482.[7] A. Itaí, C.H. Papadimitriou, J.L. Szwarcfiter, Hamilton paths in grid graphs, SIAM J. Comput. 11 (1982) 676–686.