EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015 Column Planarity and Partial Simultaneous Geometric Embedding for Outerplanar Graphs∗ Luis Barba† Michael Hoffmann‡ Vincent Kusters‡ e b c Abstract b f a d Given a graph G = (V, E), a set R ⊆ V is column pla- a a e b nar in G if we can assign x-coordinates to the vertices d f c f d in R such that every assignment of y-coordinates to R c e gives a partial embedding of G that can be completed 1 2 3 4 1 2 3 4 to a plane straight-line embedding of the whole graph. (a) (b) (c) This notion is strongly related to unlabeled level pla- narity. We prove that every outerplanar graph on Figure 1: A graph with column planar set R = n vertices contains a column planar set of size at {a, b, d, e} and ρ = {a 7→ 2, b 7→ 4, d 7→ 1, e 7→ 3}. least n/2. (b-c) depict completed embeddings for two different We use this result to show that every pair of outer- assignments γ : R → R. planar graphs G1 and G2 on the same set V of n ver- tices admit an (n/4)-partial simultaneous geometric embedding (PSGE): a plane straight-line embedding R, there exists an injection ρ : V → R, so that em- of G1 and a plane straight-line embedding of G2 such bedding each v ∈ V at (ρ(v), γ(v)) results in a plane that n/4 vertices are mapped to the same point in the straight-line embedding of G. If V is column pla- two drawings. This is a relaxation of the well-studied nar in G, then G is ULP. Estrella-Balderrama, Fowler notion of simultaneous geometric embedding, which is and Kobourov [5] introduced ULP graphs and char- equivalent to n-PSGE. acterized ULP trees in terms of forbidden subgraphs. Fowler and Kobourov [7] extended this characteriza- 1 Introduction tion to general ULP graphs. ULP graphs are exactly the graphs that admit a simultaneous geometric em- The notion of column planarity was originally intro- bedding with a monotone path: this was the original duced by Evans et al. [6]. Informally, given a graph motivation for studying them. G = (V, E), a set R ⊆ V is column planar in G Following the characterization of ULP graphs, Di if we can assign x-coordinates to the vertices in R Giacomo et al. [4] introduce a family of graphs called such that any assignment of y-coordinates to R gives fat caterpillars and prove that they are exactly the a partial embedding of G that can be completed to graphs G = (V, E) where V is column planar in G a plane straight-line embedding of the whole graph. (they call such graphs EAP graphs). Evans et al. [6] More formally, R is column planar in G if there exists prove near-tight bounds for column planar subsets of an injection ρ : R → R such that for all ρ-compatible trees: any tree on n vertices contains a column pla- injections γ : R → R, there exists a plane straight- nar set of size at least 14n/17 and for any > 0 line embedding of G where each v ∈ R is embedded and any sufficiently large n, there exists an n-vertex at (ρ(v), γ(v)). Injection γ is ρ-compatible if the com- tree in which every column planar subset has size at bination of ρ and γ does not embed three vertices on most (5/6 + )n. Furthermore, they show that outer- a line. See Figure 1. paths (outerplanar graphs whose weak dual is a path) Column planarity is both a generalization and a always contain a column planar subset of size at least strengthening of unlabeled level planarity (ULP).A n/2. In this paper, we prove that this bound holds graph G = (V, E) is ULP if for all injections γ : V → for general outerplanar graphs. Evans et al. [6] apply their results on column pla- ∗ M. Hoffmann and V. Kusters are partially supported by the narity to give bounds for k-partial simultaneous ge- ESF EUROCORES programme EuroGIGA, CRP GraDR and the Swiss National Science Foundation, SNF Project 20GG21- ometric embedding (k-PSGE). This problem is a re- 134306. laxation of simultaneous geometric embedding (SGE), †School of Computer Science, Carleton University, Ottawa, which was introduced by Brass et al. [3]. Given graphs Canada. D´epartement d’Informatique, Universit´e Libre de G = (V, E ) and G = (V, E ) on the same set of Bruxelles, Brussels, Belgium. [email protected] 1 1 2 2 ‡Department of Computer Science, ETH Zurich, Switzer- n vertices, an SGE of G1 and G2 is a pair of plane 2 land. straight-line embeddings ϕ1 : V → R of G1 and This is an extended abstract of a presentation given at EuroCG 2015. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal. 31st European Workshop on Computational Geometry, 2015 e b b e b b d suffices to compute a set R with |R| ≥ n/4 that is col- e umn planar in both G1 and G2. The result of Evans et f e a f al. [6] implies then that G1 and G2 admit a |R|-PSGE. d f e a a a b We first compute a column planar set R1 in G1. Next, c d f c d c d c we compute a column planar set R in G2[R1] with (a) (b) (c) (d) |R| ≥ n/4. Since R ⊆ R1, the set R is column planar in both G1 and G2, and hence the statement follows. Figure 2: (a-b) Graphs G1 and G2 on the same vertex set. (c) An SGE of G1 and G2. (d) A 3-PSGE of G1 2 Column Planarity in Outerplanar Graphs and G2. In this section we show that every outerplanar graph has a column planar subset containing at least half of R2 ϕ2 : V → of G2 such that ϕ1 = ϕ2. See Fig- its vertices. Let G = (V, E) be an outerplanar graph ure 2c. Conversely, in a k-PSGE of G1 and G2, we with n vertices. Assume without loss of generality require ϕ1(v) = ϕ2(v) only for some k vertices in V . that G is maximal outerplanar. More formally, a k-PSGE of G1 and G2 is a pair of Let v , v , . , v be the sequence of vertices of R2 R2 0 1 n−1 injections ϕ1 : V → and ϕ2 : V → such that V along the unique Hamiltonian cycle of G. Con- (i) the straight-line drawings ϕ1(G1) and ϕ2(G2) are sider the following removal procedure: Choose an ar- both plane; (ii) if ϕ1(v1) = ϕ2(v2) then v1 = v2; and bitrary vertex of G of degree two different from v0 (iii) ϕ1(v) = ϕ2(v) for at least k vertices v ∈ V [6]. and vn−1, remove it from the graph and repeat re- See Figure 2d. An n-PSGE is simply an SGE. cursively. Since every maximal outerplanar graph has Brass et al. [3] show that two paths, cycles, or cater- two nonadjacent vertices of degree 2, and since remov- pillars always admit an SGE. On the negative side, ing such a vertex maintains maximal outerplanarity, they prove that two outerplanar graphs or three paths such a vertex always exists. The removal order of the sometimes do not admit an SGE. Bl¨asius et al. [2] give the vertices V \{v0, vn−1} is the order in which they an excellent survey of the subsequent papers on simul- are removed by this procedure. For 0 ≤ i < n, let taneous embeddings. We highlight the negative result by Geyer et al. [8] that there exist two trees that do V (vi) = {vj ∈ V : vj was removed before vi}. not admit an SGE and the result by Angelini et al. [1] Let N +(v ) be the closed neighborhood of v . For that there exist a tree and a path that do not admit an i i 0 < i < n − 1, the left index ` of v is the smallest SGE. These negative results motivated the study of i i index such that v ∈ N +(v ). Similarly, the right PSGE. Evans et al. [6] show that if a set R is column `i i index r of v is the largest index with v ∈ N +(v ). planar in both G and G , then G and G admit a i i ri i 1 2 1 2 Naturally, v ≤ v ≤ v . |R|-PSGE. Di Giacomo et al. [4] independently prove `i i ri this for R = V . Combining their lower bounds on Lemma 1 Let vi be a vertex with 0 < i < n − 1 the size of column planar sets with a pigeonhole ar- and suppose that there is a vertex vj with i 6= j and gument, Evans et al. show that every two trees admit `i < j < ri. Then all neighbors of vj are in V (vi). a (11/17)-PSGE. A result from Goaoc et al. [9] on the untangling of Proof. Let ` = `i and r = ri and assume without outerplanar graphs, implies that any two outerplanar loss of generality that i < j. Since i < r − 1, the edge p v v is a chord of G. See Figure 3. Hence, the removal graphs G1 and G2 on n vertices admit a n/2-PSGE.