Author:
Chaplick Steven,Da Lozzo Giordano,Di Giacomo Emilio,Liotta Giuseppe,Montecchiani Fabrizio
Abstract
AbstractThe planar slope number$${{\,\textrm{psn}\,}}(G)$$
psn
(
G
)
of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that $${{\,\textrm{psn}\,}}(G) \in O(c^{\Delta })$$
psn
(
G
)
∈
O
(
c
Δ
)
for every planar graph G of maximum degree $$\Delta $$
Δ
. This upper bound has been improved to $$O(\Delta ^5)$$
O
(
Δ
5
)
if G has treewidth three, and to $$O(\Delta )$$
O
(
Δ
)
if G has treewidth two. In this paper we prove $${{\,\textrm{psn}\,}}(G) \le \max \{4,\Delta \}$$
psn
(
G
)
≤
max
{
4
,
Δ
}
when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $$O(\Delta ^2)$$
O
(
Δ
2
)
slopes suffice for nested pseudotrees.
Funder
Università degli Studi di Perugia
Publisher
Springer Science and Business Media LLC