”A separator theorem for string graphs and its applications”
by J. Fox and J. Pach
String graph — intersection graph of curves in the plane.
γ1
v1
γ2
v2
γ3
v3
Examples: planar graphs, clique.
Weight function — w : V → R≥0 , sum of weights is at most 1. If the weight function is not
specified, w(v) = |V1 | for every v ∈ V .
Separator in a graph — a subset S ⊆ V for which there is a partition V = S ∪ V1 ∪ V2 such
that w(V1 ), w(V2 ) ≤ 23 and there are no edges between V1 and V2 .
V1
S
V2
Theorem 1 (Lipton and Tarjan, ’79) For every planar graph G with n vertices and for every
√
weight function w for G there is a separator of size O( n).
√
√
√
A grid of size n × n requires a separator of size O( n), so the result is tight.
The main result of this paper is a separator theorem for string graphs:
Theorem 2 For every string√graph G with m edges and for every weight function w for G, there
is a separator of size O(m3/4 log m) with respect to w.
Notation:
• n — number of vertices in G, m — number of edges in G
• pair-crossing number pcr(G) — minimum number of pairs of edges that intersect in a drawing
of G [an intersecting pair adds 1 to the value of pcr(G), even if it intersects more than once]
• p — number of paths of length at most 3 in G
1
• bisection width b(G) — smallest integer for which there is a partition V = V1 ∪ V2 such that
|V1 |, |V2 | ≤ 23 |V | and the number of edges between V1 and V2 is b(G)
P
• ssqd(G) = v∈V (deg(v))2 [note: ssqd(G) is twice the number of paths of length 1 or 2 in G]
• ∆ — maximum degree of G
• bisection width bw (G) with respect to a weight function w — smallest integer for which there
is a partition V = V1 ∪ V2 such that w(V1 ), w(V2 ) ≤ 23 and the number of edges between V1
and V2 is bw (G)
To prove Theorem 2, we will use the following results:
Lemma 3 If G is a string graph, then pcr(G) ≤ p.
Lemma 4 (Kolman and Matousek) Every graph G on n vertices satisfies
p
p
b(G) ≤ c log n pcr(G) + ssqd(G) ,
where c is an absolute constant.
√
Corollary 5 If G is a string graph on n vertices, then b(G) = O( p log n).
√
Theorem 6 Every unweighted string graph has a separator of size O(∆ m log m).
Theorem 7 (unweighted
version of Theorem 2) Every unweighted string graph has a sepa√
rator of size O(m3/4 log m).
Theorem 8 Let G be a graph with maximum degree d, and assume that it has a drawing where
every edge intersects at most D other edges. For any weight function we have
√
√
bw (G) = O(( dD + d) n log n).
√
Theorem 9 Every weighted string graph has a separator of size O(∆ m log m) with respect to w.
2