Boolean dimension and local dimension

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x<y$?". This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.

[1]  Sean McGuinness.  On bounding the chromatic number of L-graphs , 1996, Discret. Math..

[2]  Bartosz Walczak,et al.  Minors and dimension , 2014, J. Comb. Theory, Ser. B.

[3]  William T. Trotter,et al.  On the Dimension of Posets with Cover Graphs of Treewidth 2 , 2017, Order.

[4]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[5]  Stefan Felsner,et al.  Adjacency posets of planar graphs , 2010, Discret. Math..

[6]  Roger C. Entringer,et al.  Arc Colorings of Digraphs , 1972 .

[7]  Veit Wiechert,et al.  Sparsity and dimension , 2016, SODA 2016.

[8]  Oliver Pretzel,et al.  On the Dimension of Partially Ordered Sets , 1977, J. Comb. Theory, Ser. A.

[9]  William T. TrotterJr..  Order preserving embeddings of aographs , 1978 .

[10]  Csaba Biró,et al.  Posets with Cover Graph of Pathwidth two have Bounded Dimension , 2016, Order.

[11]  Peter C. Fishburn,et al.  Partial orders of dimension 2 , 1972, Networks.

[12]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .

[13]  William T. Trotter,et al.  Dimension and height for posets with planar cover graphs , 2011, Eur. J. Comb..

[14]  William T. Trotter,et al.  The dimension of planar posets , 1977, J. Comb. Theory, Ser. B.

[15]  Kolja B. Knauer,et al.  Three ways to cover a graph , 2012, Discret. Math..

[16]  Paolo Giulio Franciosa,et al.  On the Boolean dimension of spherical orders , 1996 .

[17]  Raphaël Clifford,et al.  ACM-SIAM Symposium on Discrete Algorithms , 2015, SODA 2015.

[18]  William T. Trotter,et al.  Tree-width and dimension , 2016, Comb..

[19]  Jaroslav Nesetril,et al.  On Locally Presented Posets , 1990, Theor. Comput. Sci..

[20]  John B. Kelly,et al.  PATHS AND CIRCUITS IN CRITICAL GRAPHS. , 1954 .

[21]  Thomas Bläsius,et al.  Local and Union Boxicity , 2018, Discret. Math..

[22]  T. Hiraguchi.  On the Dimension of Partially Ordered Sets. , 1951 .

[23]  Pavel Pudlák,et al.  A Note on Boolean Dimension of Posets , 1989 .