Abstract
The general consecutive one’s property is applied to matrices with prescribed rows corresponding to sets of elements from a specified set. In applications, the rows and the columns often correspond to vertices of a graph and the entries are determined by the existence of an edge (1) between two vertices or nonexistence of the edge (0) between the two vertices. A 1 on the diagonal implies a loop at the vertex. Similarly, if the matrix represents a relation on a set, all 1’s on the diagonal corresponds to a reflexive relation and all 0’s on the diagonal corresponds to in irreflexive relation. Often in applications we don’t know if the relation is reflexive or irreflexive, nor do we care. We don’t care if an element is related to itself or not. What we are interested in is a linear ordering of the vertices (elements) such that the neighbors of an element appear consecutively in the ordering with or without the vertex itself. We present the basics of an algorithm that determines in linear time if such an ordering is possible. This algorithm determines when a diagonal element must be 0 and when it must be 1, and if an ordering is possible, produces the ordering. Roberts showed in 1968 that if all of the diagonal entries are 1 then the corresponding graph (with loops assumed at each vertex) is an indifference graph. In 1970, Tucker provided a characterization of 0 – 1 matrices with the consecutive ones property for columns. This characterization can be used for a prescribed set of diagonal entries, but would require 2n applications to check all possible sets of diagonal entries for the adjacency matrix of an n vertex graph.
This work applies in the social sciences where relationships are “to” a particular individual, country, product, etc.. For example, we might want a linear ordering of countries such that all those countries that are allies of the US appear together in the order, all those allies of Russia appear together in the order, etc., and we don’t care if, in the list of allies of the US, the US appears or not. The class of graphs whose corresponding adjacency matrix has the consecutive ones property for some choice of diagonal elements is a class of perfectly orderable (thus perfect) graphs. This class neither contains nor is contained in any of the other known classes of perfectly orderable graphs.
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References
J.J. Bartholdi, III, J.B. Orlin and H.D. Ratliff, Circular ones and cyclic staffing, in Tech. Report No. 21, Dept. of Oper. Res., Stanford Univ., 1977.
K.S. Booth and S.G. Lueker, Testing for Consecutive ones property, interval graphs, and planarity using PQ-tree algorithms, J. Comput. Syst. Sci., 13 (1976), pp. 335–379.
V. Chvàtal, Perfectly ordered graphs, in Topics on Perfect Graphs, C. Berge and V. Chvàtal Eds, North Holland, Amsterdam, 1984, pp. 63–65.
V. Chvàtal, C.T. Hoàng, N.V.R. Mahadev and D. de Werra, Four Classes of Perfectly Orderahle Graphs, J. of Graph Theory, 11, No. 4 (1987), pp. 481–495.
C. Coombs and J. Smith, On the detection of structures in attitude and developmental processes, Psych. Rev., 80 (1973), pp. 337–351.
M.B. Cozzens and F.S. Roberts, T-colorings of graphs and the channel assignment problem, Congressus Numeratum, 35 (1982).
M.B. Cozzens and R. Leibowitz, Threshold dimension of graphs, SIAM J. of Alg. and Disc. Meth., 5, No. 4 (1984).
M.B. Cozzens and N.V.R. Mahadev, Recognizing mixed diagonal consecutive ones graphs, to be submitted to, SIAM J. of Discrete Math.
S. Földes and P.L. Hammer, The Dilworth number, Ann Discrete Math., 2 (1978), pp. 211–219.
D.R. Fulkerson and O.A. Gross, Incidence matrices and interval graphs, Pacific J. Math., 15 (1965), pp. 835–855.
S.P. Ghosh, File organization: the consecutive retrieval property, Comm. Assoc. Comput. Mach., 15 (1972), pp. 802–808.
—, On the theory of consecutive storage of relevant records, J. Inform. Sci., 6 (1973), pp. 1–9.
—, File organization: consecutive storage of relevant records on a drum-type storage, Inform. Control, 25 (1974), pp. 145–165.
—, Consecutive storage of relevant records with redundancy, Comm. Assoc. Comput. Mach., 18 (1975), pp. 464–471.
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
U. Gupta, Bounds on storage for consecutive retrieval, J. Assoc. Comput. Mach., 26 (1979), pp. 28–36.
P.L. Hammer and N.V.R. Mahadev, Bithreshold graphs, SIAM J. Discrete Math., 6 (1985), pp. 497–506.
D.G. Kendall, Incidence matrices, interval graphs, and seriation in archeology, Pac. J. Math., 28 (1969), pp. 565–570.
W. Lipski, Jr. and T. Nakano, A note on the consecutive Vs property (infinite case), Comment. Math. Univ. St. Paul, 25 (1976/1977), pp. 149–152.
R.D. Luce, Semiorders and a theory of utility discrimination, Econometrica, 24 (1956), pp. 178–191.
J.R. Lundgren and J.S. Maybee, Food webs with interval competition graphs, in Graphs and Applications: Proceedings of the First Colorado Symposium on Graph Theory, Wiley, New York, 1984.
D.W. Matula, A min-max theorem with application to graph coloring, SIAM Rev., 10 (1968), pp. 481–482.
T. Nakano, A characterization of intervals; the consecutive (one’s or retrieval) property, Comment. Math. Univ. St. Paul, 22 (1973a), pp. 49–59.
—, A remark on the consecutivity of incidence matrices, Comment. Math. Univ. St. Paul, 22 (1973b), pp. 61–62.
R. Opsut and F. S. Roberts, I-colorings, I-phasings, and I-interval assignments for graphs, and their applications, Networks (1983).
On the fleet maintenance, mobile radio frequency, task assignments, and traffic phasing problems, in Proceedings of the Fourth International Conference on Theory and Applications of Graphs, Wiley, New York, 1980.
A.N. Patrinos and S.L. Hakimi, File organization with consecutive retrieval and related properties, in Large Scale Dynamical Systems, R. Sacks, ed., Point Lobos, North Hollywood, California, 1976.
M. Preissmann, D. de Werra and N.V.R. Mahadev, A Note on Superbrittle Graphs, Discrete Mathematics, 61 (1986), pp. 259–267.
F.S. Roberts, Representations of indifference relations, Ph.D. thesis, Stanford Univ., 1968.
—, Indifference graphs, in Proof Techniques in Graph Theory, F. Harary, ed., Academic Press, New York, 1969, pp. 139–146.
—, GraphTheory and its Applications to Problems of Society, NSF-CBMS Monograph, #29, SIAM Publications, Philadelphia, 1978.
—, Indifference and seriation, in Advances in Graph Theory, F. Harary (ed.), Proc. NY Acad. of Sciences, 1979, pp. -999.
—, Applications of the theory of meaningfulness to order and matching experiments, in Trends in Mathematical Psychology, E. DeGreef and J. van Buggenhaut (eds.), North Holland, Amsterdam, 1984, pp. 283–292.
D. Skrein, A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular -arc graphs, and nested interval graphs, Journal of Graph Theory, 6 (1982), pp. 309–316.
K.E. Stoffers, Scheduling of traffic lights — a new approach, Transport Res., 2 (1968), pp. 199–234.
A.C. Tucker, Characterizing the consecutive 1’s property,, in Proc. 2nd Chapel Hill Conf. on Combinatorial Mathematics and its Applications, Univ. North Carolina, Chapel Hill, 1970a, pp. 472–477.
—, A structure theorem for the consecutive 1’s property, J. Combin. Theory, 12 (1972), pp. 153–195.
A. Waksman and M.W. Green, On the consecutive retrieval property in file organization, IEEE Trans. Comput, C-23 (1974), pp. 173–174.
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Cozzens, M.B., Mahadev, N.V.R. (1989). Consecutive One’s Properties for Matrices and Graphs Including Variable Diagonal Entries. In: Roberts, F. (eds) Applications of Combinatorics and Graph Theory to the Biological and Social Sciences. The IMA Volumes in Mathematics and Its Applications, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-6381-1_3
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DOI: https://doi.org/10.1007/978-1-4684-6381-1_3
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