Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
The Compactness of Interval Routing
SIAM Journal on Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Linear algorithms to recognize interval graphs and test for the consecutive ones property
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Extremal Graph Theory
On the hardness of minimizing space for all-shortest-path interval routing schemes
Theoretical Computer Science
A new characterization of matrices with the consecutive ones property
Discrete Applied Mathematics
(Nearly-)tight bounds on the contiguity and linearity of cographs
Theoretical Computer Science
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We show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G)=5 has a Hamiltonian circuit if and only if the matrix A+I can be permuted on rows such that each column has at most (or exactly) k-1 circular blocks of consecutive ones; and if the graph G is k-regular except for two (k-1)-degree vertices a and b, then there is a Hamiltonian path from a to b if and only if the matrix A+I can be permuted on rows to have at most (or exactly) k-1 linear blocks per column. Then we turn to the problem of determining whether a given matrix can have at most k blocks of consecutive ones per column by some row permutation. For this problem, Booth and Lueker gave a linear algorithm for k=1 [Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, 1975, pp. 255-265]; Flammini et al. showed its NP-completeness for general k [Algorithmica 16 (1996) 549-568]; and Goldberg et al. proved the same for every fixed k=2 [J. Comput. Biol. 2 (1) (1995) 139-152]. In this paper, we strengthen their result by proving that the problem remains NP-complete for every constant k=2 even if the matrix is restricted to (1) symmetric, or (2) having at most three blocks per row.