Recognition and Drawing of Stick Graphs
Felice De Luca, Md Iqbal Hossain, Stephen Kobourov, Anna Lubiw and, Debajyoti Mondal

TL;DR
This paper characterizes Stick graphs through forbidden submatrices, provides algorithms for testing and constructing Stick representations given certain orderings, and explores the complexity of recognizing such graphs.
Contribution
It offers a matrix characterization of Stick graphs, an efficient algorithm for fixed orderings, and partial results on recognition complexity without orderings.
Findings
Characterization of Stick graphs via forbidden submatrices
Linear-time algorithm for fixed vertex orderings
Partial results on recognition complexity without orderings
Abstract
A \emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line,' a line with slope . It is an open question to decide in polynomial time whether a given bipartite graph with bipartition has a Stick representation where the vertices in and correspond to horizontal and vertical segments, respectively. We prove that has a Stick representation if and only if there are orderings of and such that 's bipartite adjacency matrix with rows and columns excludes three small `forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of and permit a Stick representation respecting those orderings,…
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