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Research problems
Dkrete Mathematics 44 (1983) 231-233 North-Holland Publishing Company
With Volume 36 of Discrete Mathematics, a Research Problem Section has been established. Problems in this section are intended to be research level problems rather than standard exercises. People wishing to submit such problems should send them (in duplicate) to:
Professor Brian Alspach, Department of AMathematics, Simon Fraser Unkersity, Burnaby, B.C. VSA lS6? Canada. The following should bz included: (1) The name of the person(s) who originally posed the problem; (2) the name and ;?ddress of a person willing to act as a correspondent: and (3) references and otllrer pertinent information. The Editorial Board of Discrete Mathematics invites reader& to provide information about solutions, partial resu!ts and other Pertinent items related to problems posed earlier, if possible indicating the sdurce of the information, for example papers appearing in dit%rent journals, preprints, etc. This information will be passed along to readers ‘ram time to time in order to keep them appraised of the current status of varicus problems. People wishing to provi.te information about problems that appeared earlier should write to Professcr Alspach. People wishing to correspond on technical matters concerning a problem should write to the correspondent.
Problem 35. Posed by Aviezri S. Fraenkel, Johnson. Correspondent:
Michael R. Garey
and David S.
Aviezri S. Fraenkel, Department of Applied Mathematics, The Weizm=lnn Institute of Science, Rehovot 76100, Israel.
A number of board games have recently been proved Pspace-hard on pt x n boards. Among them: checkers [I], go [3], gobang [Sj, hex [6] and press-ups [4]. 0012-3651W83/OOOO-0000/$03.00 @ 1983 North-MoLland
Rlesemh problems
232
All these proofs use reductions from planar versions of the ‘geography game’ [7]. Nevertheless, the proofs are quite dissimilar, some of them quite intricate. Is there a single Pspace-complete problem from which the various above results can be deduced easily and uniformly? This problem may be an ‘artificial’ one, analogously to Lichtenstein’s planar satisfiability 121, from which many planar graph-theoretic NP-completeness resu%s have been deduced easily and uniformly, though others have been deduced ‘by specialized constructions.
[ 11 A.S. Fraenke:. M.R. Garey. D.S. Johnson. ‘F. Schaefer and Y. Yesha, The complexity of checkers on an n x n board-preliminary report, Proc. 19th Annual Symp. Foundations Computer Science (IEEE Computer Sot., 1978) 55-64. [2] D. Lichenstein. Planar formulae and their uses, SIAM J. Comput. 11 (1982) 329-343. [f] D. Lichtenstein and M. Sipser. Go is polynomial-space hard, J. Assoc. Comput. Mach. 27 (1980) 39330 1. [-#J G.L. Peterson. Press-ups is Pspace-complete, preprint. 151 S. Reisch, Gohang ist PSPACE-vo’llstlndig, Acta Informat. 13 (1980) 59-66. 161 S. Reisch. Hex ist PSPACE-vollstindip, .4cta Informat. 15 (1981) 167-191. 171 ‘F.J. Schaefer. Gn the complexity of some two-person perfect-information games, J. Comput. Systems Sci. 16 ( 1978) 185-225.
Problem 36. Posed by Aviezri S. Fraenkel. Correspondent:
Aviezri S. Fraen kel, Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel.
Many people have recently askeld whether go on an n x n board is Exptimecomplete. See, e.g. Stockmeyer and Chandra [3]. Using a method similar to the proof of the Exptime-completeness of n x n chess [ 11, J.M. Robson showed that even checkers is Exptime-complete [2]. (Of course n x n hex is probably not Exptirle-complete, since it is Pspace-complete.) For attempting to show that go is Exptime-complete, John Reif, M. Abramski snd &ers suggest to use the Ko rule and ccnstruct a reduction from one of the Exptime-complete Boolearr games of [3]. In another direction we can ask, analogously to Problem 35, whether there is a single Exptime-complete problem from which the Lxptime-completeness of chess, checkers and other games can be deduced easily and uniformly. References [ 1J A.S. Fraenkel and D. Lichtenstein, Computing a perfect strategy for n x n chess expclnentialin n. J. Combin. Theory (AI 31 (1981) 199-214.
requires time
Re; zarch problems
233
PI J.M. Robson, N by N checkers is Exptinw-compiete, Australian National University, Department of Computer Science, preprint.
131 L.J. Stockmeyer and A. K. Chandra, Provably difIicult combinatorial games, SIAM J. Comput. 8 (1979) 15 1-174.
Problem 37. Posed by D. Holton and M.D. Plummer. Correspondent:
M.D. Plummer, Department of Mathematics, Vanderbilt University, Nashville, TN 37235, U.S.A.
Let X be any 2-connected graph. Chvatal has defined the cyclabili~ of X to be the minimum m such that every set of m vertices lie in a cycle of X. If k a 2, denote by f(k) the minimum cyclability taken over all k-regular, k-connected graphs. It has been proved in [ 1] and [3] that f(k) 2 k + 4. More particularly, it has been shown in [2] and [3] that f(3) = 9. A family of graphs constructed in [4] serves to establish that f(k) G 10k - 11. The problem is to determine better (or best!) bounds for f(k), k 3 4. References D.A. Hdton, Cycles through specified vertices in k-connected regular graphs. Ars Combin. 13 (1982) 129-144. D.A. Holton. B.D. McKay, M.D. Plummer and C. Thomassen, Cycles through specified vertices in 3-connected cubic graphs, Combinatorics 2 ( 1982) to appear. A.K. Kelmans and M.V. Lomonosov, When nt vertices in a &-connected graph cannot be walked . around along a simple cycle. Discrete Math. 38 (1982) 317-322. G.H.J. Meredith, Regular n-valent n-connected nonhamiltonian non-n-edge-colorable graphs, J. Combin. Theory (B) 14 (1973) 55-60.
With Volume 36 of Discrete Mathematics, a Research Problem Section has been established. Problems in this section are intended to be research level problems rather than standard exercises. People wishing to submit such problems should send them (in duplicate) to:
Professor Brian Alspach, Department of AMathematics, Simon Fraser Unkersity, Burnaby, B.C. VSA lS6? Canada. The following should bz included: (1) The name of the person(s) who originally posed the problem; (2) the name and ;?ddress of a person willing to act as a correspondent: and (3) references and otllrer pertinent information. The Editorial Board of Discrete Mathematics invites reader& to provide information about solutions, partial resu!ts and other Pertinent items related to problems posed earlier, if possible indicating the sdurce of the information, for example papers appearing in dit%rent journals, preprints, etc. This information will be passed along to readers ‘ram time to time in order to keep them appraised of the current status of varicus problems. People wishing to provi.te information about problems that appeared earlier should write to Professcr Alspach. People wishing to correspond on technical matters concerning a problem should write to the correspondent.
Problem 35. Posed by Aviezri S. Fraenkel, Johnson. Correspondent:
Michael R. Garey
and David S.
Aviezri S. Fraenkel, Department of Applied Mathematics, The Weizm=lnn Institute of Science, Rehovot 76100, Israel.
A number of board games have recently been proved Pspace-hard on pt x n boards. Among them: checkers [I], go [3], gobang [Sj, hex [6] and press-ups [4]. 0012-3651W83/OOOO-0000/$03.00 @ 1983 North-MoLland
Rlesemh problems
232
All these proofs use reductions from planar versions of the ‘geography game’ [7]. Nevertheless, the proofs are quite dissimilar, some of them quite intricate. Is there a single Pspace-complete problem from which the various above results can be deduced easily and uniformly? This problem may be an ‘artificial’ one, analogously to Lichtenstein’s planar satisfiability 121, from which many planar graph-theoretic NP-completeness resu%s have been deduced easily and uniformly, though others have been deduced ‘by specialized constructions.
[ 11 A.S. Fraenke:. M.R. Garey. D.S. Johnson. ‘F. Schaefer and Y. Yesha, The complexity of checkers on an n x n board-preliminary report, Proc. 19th Annual Symp. Foundations Computer Science (IEEE Computer Sot., 1978) 55-64. [2] D. Lichenstein. Planar formulae and their uses, SIAM J. Comput. 11 (1982) 329-343. [f] D. Lichtenstein and M. Sipser. Go is polynomial-space hard, J. Assoc. Comput. Mach. 27 (1980) 39330 1. [-#J G.L. Peterson. Press-ups is Pspace-complete, preprint. 151 S. Reisch, Gohang ist PSPACE-vo’llstlndig, Acta Informat. 13 (1980) 59-66. 161 S. Reisch. Hex ist PSPACE-vollstindip, .4cta Informat. 15 (1981) 167-191. 171 ‘F.J. Schaefer. Gn the complexity of some two-person perfect-information games, J. Comput. Systems Sci. 16 ( 1978) 185-225.
Problem 36. Posed by Aviezri S. Fraenkel. Correspondent:
Aviezri S. Fraen kel, Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel.
Many people have recently askeld whether go on an n x n board is Exptimecomplete. See, e.g. Stockmeyer and Chandra [3]. Using a method similar to the proof of the Exptime-completeness of n x n chess [ 11, J.M. Robson showed that even checkers is Exptime-complete [2]. (Of course n x n hex is probably not Exptirle-complete, since it is Pspace-complete.) For attempting to show that go is Exptime-complete, John Reif, M. Abramski snd &ers suggest to use the Ko rule and ccnstruct a reduction from one of the Exptime-complete Boolearr games of [3]. In another direction we can ask, analogously to Problem 35, whether there is a single Exptime-complete problem from which the Lxptime-completeness of chess, checkers and other games can be deduced easily and uniformly. References [ 1J A.S. Fraenkel and D. Lichtenstein, Computing a perfect strategy for n x n chess expclnentialin n. J. Combin. Theory (AI 31 (1981) 199-214.
requires time
Re; zarch problems
233
PI J.M. Robson, N by N checkers is Exptinw-compiete, Australian National University, Department of Computer Science, preprint.
131 L.J. Stockmeyer and A. K. Chandra, Provably difIicult combinatorial games, SIAM J. Comput. 8 (1979) 15 1-174.
Problem 37. Posed by D. Holton and M.D. Plummer. Correspondent:
M.D. Plummer, Department of Mathematics, Vanderbilt University, Nashville, TN 37235, U.S.A.
Let X be any 2-connected graph. Chvatal has defined the cyclabili~ of X to be the minimum m such that every set of m vertices lie in a cycle of X. If k a 2, denote by f(k) the minimum cyclability taken over all k-regular, k-connected graphs. It has been proved in [ 1] and [3] that f(k) 2 k + 4. More particularly, it has been shown in [2] and [3] that f(3) = 9. A family of graphs constructed in [4] serves to establish that f(k) G 10k - 11. The problem is to determine better (or best!) bounds for f(k), k 3 4. References D.A. Hdton, Cycles through specified vertices in k-connected regular graphs. Ars Combin. 13 (1982) 129-144. D.A. Holton. B.D. McKay, M.D. Plummer and C. Thomassen, Cycles through specified vertices in 3-connected cubic graphs, Combinatorics 2 ( 1982) to appear. A.K. Kelmans and M.V. Lomonosov, When nt vertices in a &-connected graph cannot be walked . around along a simple cycle. Discrete Math. 38 (1982) 317-322. G.H.J. Meredith, Regular n-valent n-connected nonhamiltonian non-n-edge-colorable graphs, J. Combin. Theory (B) 14 (1973) 55-60.