A 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 −1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G’s bipartite adjacency matrix with rows A and columns B 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 A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A|3|B|3)-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.
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Recognition and Drawing of Stick Graphs
A 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 −1. It is an open question to decide in polynomial time
whether a given bipartite graph G with bipartition A∪B has a Stick representation
where the vertices in A and B correspond to horizontal and
vertical segments, respectively. We prove that G has a Stick representation
if and only if there are orderings of A and B such that G’s bipartite
adjacency matrix with rows A and columns B 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 A and B permit a Stick representation respecting those orderings, and to find such
a representation if it exists. The algorithm runs in time linear in the size
of the adjacency matrix. For the case when only the ordering of A is
given, we present an O(|AB}^3)-time algorithm.
more »
« less
- Award ID(s):
- 1712119
- NSF-PAR ID:
- 10065741
- Date Published:
- Journal Name:
- 26th Symposium on Graph Drawing (GD)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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