A tanglegram Kuratowski theorem

Abstract

A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing, a tanglegram is called planar. We show that every nonplanar tanglegram contains one of two nonplanar 4‐leaf tanglegrams as an induced subtanglegram, which parallels Kuratowski's Theorem.

Analogies between the Crossing Number and the Tangle Crossing Number

“…Throughout this section, given a tanglegram T = (L, R, M ) and e ∈ M , we use T − e to denote the subtanglegram of T induced by edges in M − e. Proof. We will proceed by induction on n. First observe that if T is a tanglegram of size at most 3 then it is planar, and if T is a tanglegram of size 4 then crt(T ) ≤ 1 [6]; so the theorem is trivial when n ≤ 4. Let n ≥ 5 and suppose that in every tanglegram of size n − 1, each edge contributes at most (n − 1) − 3 to the tangle crossing number.…”

“…Analogously, testing for tangle crossing number 0 can also be done in linear time [8]. Recently, Czabarka, Székely, and Wagner [6] gave an analogue of Kuratowski's Theorem [13] for tanglegrams, characterizing tangle crossing number 0. Clearly, for a graph G with e edges we have cr(G) = O(e 2 ), while for a tanglegram T of size n, crt(T ) = O(n 2 ).…”

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

“…Some of the known results in the Graph Drawing Problem have analogous results in tanglegram layouts, and some have approached the Tanglegram Layout Problem by translating what we know about graphs to tanglegrams. Czabarka, Székely, and Wagner recently used the well-known Kuratowski Theorem characterizing planarity of graphs to construct a Tanglegram Kuratowski Theorem characterizing planar tanglegrams, which are tanglegrams with crossing number zero [5]. Prior to this, Lozano et.…”

“…In this paper, we will explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for drawing a planar layout [5,16,15]. We start by building on these results and characterizing all planar layouts of a planar tanglegram.…”

“…For k = 1, consider the tanglegram (T, S, φ) with two layouts in Figure 18. Direct application of ModifiedUntangle or the Tanglegram Kuratowski Theorem in [5] shows that the induced subtanglegram on [n] \ {i} is planar only when i = 2, and hence crtei(T, S, φ) = crtei((T, S, φ), 2). Insertion((T, S, φ), 2) outputs the first layout in Figure 18 with three crossings, which is a solution to the Tanglegram Single Edge Insertion Problem by Theorem 1.3.…”