2022/02/24 by Mathew C. Francis, Francis, Mathew C., Atrayee Majumder +3
Computer Science · Mathematics · #05C31 #05C62 #05C69 #05C80 #05D40 #94C11 #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2202.12325
openalex publication_date 2022/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∑ i=1n ai xi ≤ 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\colon \0,1\n → \0,1\ is the Boolean function that has the property that fG(x) = 1 if and only if x is the characteristic vector of a clique in G. A Boolean function f for which there exists a graph G such that f=fG is called a graphic Boolean function. It follows that for a graph G, \dimth(G) is precisely the minimum number of majority gates whose AND (or conjunction) realizes the graphic Boolean function fG. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth.