The Unit Bar Visibility Number of a Graph

A t-unit-bar representation of a graph G is an assignment of sets of at most t horizontal unit-length segments in the plane to the vertices of G so that (1) all of the segments are pairwise nonintersecting, and (2) two vertices x and y are adjacent if and only if there is a vertical channel of positive width connecting a segment assigned to x and a segment assigned to y that intersects no other segment. The unit bar visibility number of a graph G, denoted ub(G), is the minimum t such that G has a t-unit-bar visibility representation. Our results include a linear time algorithm that determines ub(T ) when T is a tree, bounds on ub(Km,n) that determine ub(Km,n) asymptotically when n and m are asymptotically equal, and bounds on ub(Kn) that determine ub(Kn) exactly when n ≡ 1, 2 (mod 6). Submitted: September 2015 Reviewed: December 2015 Revised: February 2016 Accepted: March 2016 Final: March 2016 Published: April 2016 Article type: Regular paper Communicated by: W. S. Evans Research supported by REU in Extremal Graph Theory and Dynamical Systems – NSF award