1 research outputs found
On String Contact Representations in 3D
An axis-aligned string is a simple polygonal path, where each line segment is
parallel to an axis in . Given a graph , a string contact
representation of maps the vertices of to interior disjoint
axis-aligned strings, where no three strings meet at a point, and two strings
share a common point if and only if their corresponding vertices are adjacent
in . The complexity of is the minimum integer such that every
string in is a -string, i.e., a string with at most bends.
While a result of Duncan et al. implies that every graph with maximum
degree 4 has a string contact representation using -strings, we examine
constraints on that allow string contact representations with complexity 3,
2 or 1. We prove that if is Hamiltonian and triangle-free, then admits
a contact representation where all the strings but one are -strings. If
is 3-regular and bipartite, then admits a contact representation with
string complexity 2, and if we further restrict to be Hamiltonian, then
has a contact representation, where all the strings but one are -strings
(i.e., -shapes). Finally, we prove some complementary lower bounds on the
complexity of string contact representations