English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist realnumbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of thelinear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristicvectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals ofDiscrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$,denoted by $\dimth(G)$, is the minimum number of threshold graphs whoseintersection yields $G$. Given a graph $G$ on $n$ vertices, in line withChv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is theBoolean function that has the property that $f_G(x) = 1$ if and only if $x$ isthe characteristic vector of a clique in $G$. A Boolean function $f$ for whichthere exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Booleanfunction. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimumnumber of \emph{majority} gates whose AND (or conjunction) realizes the graphicBoolean function $f_G$. The fact that there exist Boolean functions which canbe realized as the AND of only exponentially many majority gates motivates usto study threshold dimension of graphs. We give tight or nearly tight upperbounds 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 highgirth.

Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates