Andrea S. LaPaugh
Abstract
This paper is concerned with a game on graphs called graph searching. The object of this game is to clear all edges of a contaminated graph. Clearing is achieved by moving searchers, a kind of token, along the edges of the graph according to clearing rules. Certain search strategies cause edges that have been cleared to become contaminated again. Megiddo et al. [9] conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring, that is, without clearing any edge twice. In this paper, this conjecture is proved. This places the graph-searching problem in NP, completing the proof by Megiddo et al. that the graph-searching problem is NP-complete. Furthermore, by eliminating the need to consider recontamination, this result simplifies the analysis of searcher requirements with respect to other properties of graphs.
References
- 1 ~BIENSTOCK, D., AND SEYMOUR, P. Monotonicity m graph searching. J. Algorithms 12 (1991), ~239-245. Google Scholar
- 2 ~BRElSCH, R. An intuitive approach to speleotopology. Southwestern Cat:ers (A publication of ~the Southwestern Region of the National Speleological Society) VI, 5 (Dec. 1967), 72-78.Google Scholar
- 3 ~ELLIS, J. A., SUDBOROUGH, I. H., AND TURNER, J.S. Graph separation and search number. ~In Proceedings of the Allerton Conference on Communication, Control and Compuung. Coordi- ~nated Sci. Lab., Univ. Illinois, Urbana-Champaign, I11., 1983, pp. 224 233.Google Scholar
- 4 ~GAREY, M. R., AND JOHNSON, D.S. Compttlers and bltractabiliO': A Guide to the Theoo' of ~NP-Completeness. W. H. Freeman and Co., San Francisco, Calif., 1979. Google Scholar
- 5 ~KIROUSIS, L., AND PAPADIMITRIOU, C. Searching and pebbling. Theoret. Comput. Set. 47 ~(1986), 205 218. Google Scholar
- 6 ~MAKEDON, F. Layout problems and their complexity. Ph.D. dissertation. Dept. of Electrical ~Engineering and Computer Science, Northwestern Univ., Chicago, Ill., 1982. Google Scholar
- 7 ~MAKEDON, F., PAPADIMITRIOU, C., AND SUDBOROUGH, I.H. Topological bandwidth. SIAM J. ~Al~: Disc. Meth. 6 (1985), 418-444.Google Scholar
- 8 ~MAKEDON, F., AND SUDBOROUGH, I.H. On minimizing width in linear layouts. Disc. Appl. ~ Math. 23 (1989), 243 265. Google Scholar
- 9 ~MEGiDDO, N., HAKIMI, S. L., GAREY, M. R., JOHNSON, D. S., AND PAPADINiITRIOU, C.H. The ~complexity of searching a graph, J. ACM 35, 1 (Jan. 1988), 18-44. Google Scholar
- 10 ~PARSONS, T.D. Pursuit-evasion in a graph. In Theory and ,4pphcattons of Graphs. Y. Alavi ~ and D. R. Lick, eds. Lecture Notes in Mathematics. Springer-Verlag, New York, 1976, pp. ~ 426-441.Google Scholar
- 11 ~PARSONS, T. D. The search number of a connectcd graph. In Proceedtngs of the 9th ~Southeastern Conference on Combinatodcs, Graph Theoly and Cotnputing. Utilitas Mathemat- ~ica Publishing, Winnipeg, Canada, 1978, pp. 549 554.Google Scholar
Index Terms
Recontamination does not help to search a graph
Reviews
Daniel P. Bovet
Lapaugh proves the truth of a conjecture raised by Megiddo et al. [1] that the edges of a graph can be cleared by an efficient one-pass algorithm without ever having to reclear an edge already cleared. This places the graph-searching problem in NP, completing the proof by the earlier authors that the problem is NP-complete. The rules of the game are the following: initially all edges are contaminated. An edge (u,v) can be cleared by sliding a pebble from u to v, while shielding u from contaminated (that is, uncleared) edges with appropriately placed pebbles (for example, by keeping a second pebble at u). An edge-clearing strategy is optimal if it minimizes the number of require d pebbles. The proof described is rather involved and requires over ten pages. A simpler proof of the same result has been published [2], thus placing me in an uncomfortable position. It is fair to say that the long refereeing process (the paper was submitted in January 1985) has limited the interest of this paper.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments