Bounding Threshold Dimension: Realizing Graphic Boolean Functions as the AND of Majority Gates

  title={Bounding Threshold Dimension: Realizing Graphic Boolean Functions as the AND of Majority Gates},
  author={Mathew C. Francis and Atrayee Majumder and Rogers Mathew},
A graph G on n vertices is a threshold graph if there exist real numbers a1, a2, . . . , an and b such that the zero-one solutions of the linear inequality n ∑ i=1 aixi ≤ b are the characteristic vectors of the cliques of G. Introduced in [Chvátal and Hammer, Annals of Discrete Mathematics, 1977], the threshold dimension of a graph G, denoted by dimTH(G), is the minimum number of threshold graphs whose intersection yields G. Given a graph G on n vertices, in line with Chvátal and Hammer, fG… 



Aggregation of inequalities in integer programming.

Representing graphs as the intersection of cographs and threshold graphs

Improved bounds are derived on the treewidth, pathwidth, chromatic number and boxicity of the graph G when it belongs to some special graph classes and several new bounds on these parameters are presented.

Cubicity, degeneracy, and crossing number

Depth-2 Threshold Circuits

  • M. Mahajan
  • Computer Science, Mathematics
  • 2019
Circuits with linear threshold functions as primitives are a natural model for computation in the brain. Small threshold circuits of depth two cannot compute most functions, but how do we prove such

Minimal scrambling sets of simple orders

Abstract : Let 2 < or = k < n be fixed integers, a family F of simple orders on an n element set is said to be k-suitable if of every k elements in the n set, each one is the largest of the k in some

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The dimension of a partial order P is the minimum number of linear orders whose intersection is P. There are efficient algorithms to test if a partial order has dimension 1 or 2. We prove that it is

Algorithmic graph theory and perfect graphs

Boxicity, Poset Dimension, and Excluded Minors

The results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $T^4$), and as the intersections of $\tfrac{15}2 t^2$ circular-arc graphs.

Local boxicity and maximum degree

Better bounds for poset dimension and boxicity

  • A. ScottD. Wood
  • Mathematics, Computer Science
    Transactions of the American Mathematical Society
  • 2019
It is proved that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$, and is within a $\log^{o( 1)}\Delta$ factor of optimal.