Doubly lexical orderings of matrices

A doubly lexical ordering of the rows and columns of any real-valued matrix is defined. This notion extends to graphs. These orderings are used to prove and unify results on several classes of matrices and graphs, including totally balanced matrices and chordal graphs. An almost-linear time doubly lexical ordering algorithm is given.

[1]  D. R. Fulkerson,et al.  On balanced matrices , 1974 .

[2]  G. Nemhauser,et al.  The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs , 1984 .

[3]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[4]  Martin Farber,et al.  Domination, independent domination, and duality in strongly chordal graphs , 1984, Discret. Appl. Math..

[5]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[6]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[7]  Peter Buneman,et al.  A characterisation of rigid circuit graphs , 1974, Discret. Math..

[8]  Catriel Beeri,et al.  On the Desirability of Acyclic Database Schemes , 1983, JACM.

[9]  Richard P. Anstee,et al.  Characterizations of Totally Balanced Matrices , 1984, J. Algorithms.

[10]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[11]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[12]  Ronald Fagin,et al.  Degrees of acyclicity for hypergraphs and relational database schemes , 1983, JACM.

[13]  Peter J. Slater,et al.  R-Domination in Graphs , 1976, J. ACM.

[14]  Richard P. Anstee,et al.  Properties of (0,1)-Matrices With Forbidden Configurations , 1980 .

[15]  András Frank,et al.  On a class of balanced hypergraphs , 1977, Discret. Math..

[16]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[17]  A. Hoffman,et al.  Totally-Balanced and Greedy Matrices , 1985 .