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Strong reducibility of powers of paths and

powers of cycles on Impartial Solitaire Clobber1

Telma Par´ aa,2, Simone Dantasb,2, Sylvain Gravierc,2

aCOPPE, Federal University of Rio de Janeiro, Brazil.

bGAN, IM, Fluminense Federal University, Brazil.

cCNRS, Institut Joseph Fourier, ERT Maths ` a Modeler, France.

Abstract

We consider the Impartial Solitaire Clobber which is a one-player combinatorial

game on graphs. The problem of determining the minimum number of remaining

stones after a sequence of moves was proved to be NP-hard for graphs in general and,

in particular, for grid graphs. This problem was studied for paths, cycles and trees,

and it was proved that, for any arrangement of stones, this number can be computed

in polinomial time. We study a more complex question related to determining the

color and the location of the remaining stones. A graph G is strongly 1-reducible

if: for any vertex v of G, for any arrangement of stones on G such that G \ v is

non-monochromatic, and for any color c, there exists a succession of moves that

yields a single stone of color c on v. We investigate this problem for powers of paths

Pr

nand for powers of cycles Cr

strongly 1-reducible; if r = 2, then Pr

n, is not strongly 1-reducible.

nand we prove that if r ≥ 3, then Pr

n(resp. Cr

n) is

Keywords: Graph theory, combinatorial games, impartial games, solitaire clobber.

1This research was supported by CAPES(Brazil)/COFECUB(France) and CNPq.

2Email: telma@cos.ufrj.br, sdantas@im.uff.br, sylvain.gravier@ujf-grenoble.fr

Electronic Notes in Discrete Mathematics 37 (2011) 177–182

1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.endm.2011.05.031

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1Introduction

Clobber is a combinatorial game which was first introduced by Albert et al. [1]

at the seminar on Algorithmic and Combinatorial Game Theory in Dagstuhl.

The rules of Clobber are: we place a stone on each vertex of a graph G, each

stone being black or white; a move consists of picking a stone and clobbering

another one of opposite color located on an adjacent vertex; the clobbered

stone is removed from the graph and it is replaced by the picked one; the goal

is to find a succession of moves that minimizes the number of remaining stones,

when no move is possible. Solitaire Clobber was introduced by Demaine et

al. [4] as one-player variant and it has the versions Partisan and Impartial.

The difference between them is that, in the Impartial Solitaire Clobber (ISC),

the player is not forced to alternate black and white stone moves.

The problem of determining the minimum number of remaining stones on

a graph was proved by Ita´ ı et al. [6] to be NP-hard for graphs in general

and, in particular, for grid graphs. This result is obtained by reducing the

NP-complete Hamiltonian path problem to this problem.

A configuration of a graph G is an arrangement of stones on G. If a

configuration has stones of only one color, then it is called monochromatic.

Otherwise, it is non-monochromatic.

The ISC was investigated by Blondel et al. [3] on paths and cycles, and

by Beaudou et al. [2] on trees, by considering the reducibility value, i.e., the

minimum number of remaining stones for a given configuration of G. If G is

a path or a cycle of size n then, for any configuration, one can compute the

reducibility value in linear time O(n). If G is a tree then, for any configuration,

it can be computed in O(n3) operations. Dorbec et al. [5] proved that, for any

non-monochromatic configuration, the reducibility value is 1 for Hamming

graphs, except for hypercubes which is at most 2. Par´ a et al. [7] proved that,

for any non-monochromatic configuration, if G is a power of path Pr

or a power of cycle Cr

power of path P2

In 2008, Dorbec et al. [5] introduced a more complex question about ISC:

for any vertex v of G, for any configuration of G (provided G \ v is non-

monochromatic), for any color c (black or white), does there exist a way 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. This question was studied for certain classes of

graphs and the state of the art of this problem was the following: all Hamming

graphs are strongly 1-reducible, except hypercubes and cartesian product of

K2with K3; all cliques of size n ≥ 3 are strongly 1-reducible; and if G is a

n, r ≥ 3

n, r ≥ 2, then the reducibility value is 1; and if G is a

nthen the reducibility value is at most 2.

T. Pará et al. / Electronic Notes in Discrete Mathematics 37 (2011) 177–182

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strongly 1-reducible graph containing at least 4 vertices, then the cartesian

product of G with any clique is strongly 1-reducible.

In this paper, we investigate the problem studied in [5] on powers of paths

Pr

strongly 1-reducible; if r = 2, then Pr

we determine a family of configurations of P2

and this improves the result given in [7] by stablishing that the exact re-

ducibility value is 2 for this family.

nand powers of cycles Cr

nand we prove that if r ≥ 3, then Pr

n, is not strongly 1-reducible. Moreover,

nfor which P2

nand Cr

nare

nis not 1-reducible

2Strong reducibility

A configuration Φ of a graph G = (V,E) is a mapping Φ :

Let c ∈ { , }, we denote by Φ(v) the color c of the stone on v. If color c =

(resp.) then ¯ c =

(resp.). Given a configuration Φ of G, if Φ(v) =

(resp.

), for all v ∈ V (G), then Φ is called monochromatic. Otherwise, Φ is

non-monochromatic. We say that a color c is rare on G if there exists a unique

vertex v ∈ V (G) such that v is labeled c. We say that (G,Φ) is k-reducible if

there exists a succession of moves that leaves at most k stones on the graph.

The reducibility value rv(G,Φ) is the smallest integer k for which (G,Φ) is

k-reducible. We say that (G,Φ) is (1,v,c)-reducible if (G,Φ) is 1-reducible on

v with color c, i.e., if there exists a way to play that yields only one stone of

color c, located on v.

For convenience, we may say that vertex v clobbers vertex u instead of

talking about the corresponding stones.

We define the distance, dG(u,v), as the number of edges in a shortest path

between u and v. Let Pn be a path. A power of path Pr

that V (Pr

Cnbe a cycle. A power of cycle Cr

E(Cr

V

→ { , }.

nis a graph such

n) = V (Pn) and E(Pr

n) =?uv : dPn(u,v) ≤ r ,u,v ∈ V (Pr

nis a graph such that V (Cr

n)?.

n)?. Let

n) = V (Cn) and

n) =?uv : dCn(u,v) ≤ r ,u,v ∈ V (Cr

Theorem 2.1 Powers of paths Pr

n, r ≥ 3, are strongly 1-reducible.

Proof Let A and B be two blocks of vertices and vibe a vertex of V (Pr

that V (Pr

of G, for any configuration of Pr

for any color c, (G,Φ) is (1,vi,c)-reducible. We observe that the theorem is

valid for n = 4, since by [5], all cliques are strongly 1-reducible; and it is easy

to verify for n = 5. Now we consider three cases.

Case (i): Let Φ of A and Φ of B be non-monochromatic (see Figure 1(a)).

First, we play on G[A∪{vi}]. We observe that Φ of G[A] is non-monochromatic.

n) such

n) = A∪{vi}∪B. We prove, by induction on n, that for any vertex vi

nsuch that Φ of Pr

n\viis non-monochromatic,

T. Pará et al. / Electronic Notes in Discrete Mathematics 37 (2011) 177–182

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Then, by induction hypothesis, G[A∪{vi}] is (1,vi,c)-reducible, for all c. Now

we play on G[B ∪ {vi}]. Since Φ of G[B] is non-monochromatic, again, by

induction hypothesis, G[B ∪ {vi}] is (1,vi,c)-reducible, for all c.

A

B

vi

x

x x

x

x

(a) B non-monochromatic.

A

x

B

v

x

xx

i

vi+1

vi-1

(b) B monochromatic.

Figure 1. Examples for Pr

n, r ≥ 3, with A non-monochromatic.

Case (ii): Let Φ of A be non-monochromatic and let Φ of B be monochro-

matic. Without loss of generality, suppose B of color white. See an example in

Figure 1(b). We first play on G[A]. If Φ of G[A\{vi−1}] is non-monochromatic,

then, by induction hypothesis, G[A] is (1,vi−1, )-reducible. Vertex vi−1clob-

bers vi+1. Now Φ of G[B] is non-monochromatic. Then, by induction hypoth-

esis, G[B∪{vi}] is (1,vi,c)-reducible. Case Φ of G[A\{vi−1}] is monochromatic

is similar by considering vertex vi−2.

Case (iii): Let Φ of A and Φ of B be monochromatic, then we have two

subcases: blocks A and B have the same color and blocks A or B have op-

posite colors. The first subcase will not be considered because Φ of G \ viis

monochromatic. In the second subcase, without loss of generality, we assume

that block A has color black, block B has color white and vihas color black

(see Figure 2).

Let c = Φ(vi). We proceed as follows: vi−1clobbers vi+2and vi+1clobbers

vi−2. Let A?= A\{vi−1} and B?= B\{vi+1}. We note that Φ of G[B?\{vi+3}]

is non-monochromatic. Then, by induction hypothesis, G[B?] is (1,vi+3, )-

reducible. We also verify that Φ of G[A?\ {vi−3}] is non-monochromatic.

Then, by induction hypothesis, G[A?] is (1,vi−3, )-reducible. The last moves

are: vi+3clobbers viand vi−3clobbers vi.

Now assume that c = Φ(vi). First, vi+1clobbers vi−2. We verify that Φ

of G[A \ {vi−1}] is non-monochromatic. Then, by induction hypothesis, G[A]

is (1,vi−1, )-reducible. Second, vi−1clobbers vi+2. Let B?= B \ {vi+1}. We

observe that Φ of G[B?\{vi+3}] is non-monochromatic and then, by induction

hypothesis, G[B?] is (1,vi+3, )-reducible. Finally, vi+3 clobbers vi and we

conclude the proof.

2

Therefore, next result is a consequence of Theorem 2.1 and Pr

n⊂ Cr

n.

Corollary 2.2 Powers of cycles Cr

n, r ≥ 3, are strongly 1-reducible.

2

T. Pará et al. / Electronic Notes in Discrete Mathematics 37 (2011) 177–182

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A

B

x

x

A’

B’

x

x

x

x

x

x

x

x

vi-3vi-2vi-1

vi +2

vi +1

vi +3

vi

Figure 2. Example for Pr

n, r ≥ 3, with monochromatic blocks A and B and c = Φ(v).

A

B

v

v

x

v

x

x

x

x

x

x

x

x

x

x

x

xx

x

i

i

i

-3

-2

-1

vi+1vi +2vi+3

vi

x

x

x

x

x

x

B’

Figure 3. Example for Pr

n, r ≥ 3, with monochromatic blocks A and B and c = Φ(v).

Now we study the strong reducibility of powers of paths Pr

note that for configurations Φ of P2

that Φ(v) ?= : if Φ =

then (P2

for any c; and if Φ =

then (P2

first vertex. In what follows, we present results concerning P2

Let H be an induced subgraph of P2

We refer to H of Figure 4, where x is a vertex already clobbered and Y and

Z are blocks of vertices.

n, r = 2. We

4with a rare color

4,Φ) is (1,v,c)-reducible, for any v and

4,Φ) is not (1,v, )-reducible when v is the

and for vertices v such

n.

nobtained after a sequence of moves.

Property 2.3 Graph H is called a forbidden subgraph of P2

sequence of moves does not yields one stone.

nbecause any

This occurs because even if we could reduce each block Y or Z to a single

stone, the distance between any vertex of block Y and any vertex of block Z

in Pnis greater than 2.

x x

Y

Z

Figure 4. Forbidden subgraph H.

Next result is based on Property 2.3 and its proof is omitted from this

paper due to space.

T. Pará et al. / Electronic Notes in Discrete Mathematics 37 (2011) 177–182

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Lemma 2.4 Let Φ be a non-monochromatic configuration of P2

monochromatic blocks A and B such that A ∪ B = V (P2

for all v ∈ A, and Φ(v) = , for all v ∈ B. Then (P2

nwith two

n) and Φ(v) =

n,Φ) is not 1-reducible.2

,

As a consequence of Lemma 2.4, P2

nis not strongly 1-reducible.

3Conclusions

In this work, we study the strong reducibility of powers of paths and powers

of cycles. We proved that Pr

sidering powers of paths P2

improves this result by exihibiting a family of configurations of P2

not 1-reducible and this stabilishes that the exact value of rv(G,Φ) is equal

to 2 for this family.

nand Cr

n, we showed in [7] that rv(P2

n, r ≥ 3, are strongly 1-reducible. Con-

n,Φ) ≤ 2. Lemma 2.4

nwhich is

References

[1] Albert, M. H., Grossman, J. P., Nowakowsky, R. J. and Wolfe, D.,

Introduction to Clobber, Integers, Journal of Combinatorial Number Theory,

5(2), A1, pp. 1–12, 2005.

An

[2] Beaudou, L., Duchˆ ene, E. and Gravier, S., A Survey about Solitaire Clobber,

http://www.isima.fr/˜beaudou/docs/survey clobber.pdf.

[3] Blondel, V.D., Kerchove, C. de, Hendrickx, J. M., and Jungers, R., Solitaire

Clobber as an Optimization Problem on Words,

Combinatorial Number Theory, 8(1), G04, pp. 1–12, 2008.

Integers, Journal of

[4] Demaine, E. D., Demaine, M. L., and Fleischer, R.,

Theoretical Computer Science, 313, pp. 325–338, 2004.

Solitaire Clobber,

[5] Dorbec, P., Duchˆ ene, E. and Gravier, S., Solitaire Clobber played on Hamming

graphs, Integers, Journal of Combinatorial Number Theory, 8(1), G03, pp.

1–21, 2008.

[6] Ita´ ı, A., Papadimitriou, C. H. and Szwarcfiter, J. L. Hamilton paths in grid

graphs, SIAM Journal of Computing, 11, pp. 676-686, 1982.

[7] Par´ a, T., Gravier, S., and Dantas, S., Impartial Solitaire Clobber played on

Powers of Paths, Eletronic Notes on Discrete Mathematics, 35, pp. 257–262,

2009.

T. Pará et al. / Electronic Notes in Discrete Mathematics 37 (2011) 177–182

182

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