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Emilio Di Giacomo, Giuseppe Liotta, Fabrizio Montecchiani Università degli Studi di Perugia 22 nd International Symposium on Graph Drawing September 2014, Würzburg, Germany Drawing outer 1-planar graphs with few slopes

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Problem definition, motivation, and results

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The (Planar) Slope Number of a Graph G minimum number of edge slopes to compute a (planar) straight-line drawing of G

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The (Planar) Slope Number of a Graph G minimum number of edge slopes to compute a (planar) straight-line drawing of G u v G

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The (Planar) Slope Number of a Graph G minimum number of edge slopes to compute a (planar) straight-line drawing of G u v G u v

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The (Planar) Slope Number of a Graph G minimum number of edge slopes to compute a (planar) straight-line drawing of G u v G u v psl(G)=5

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Given a family of (planar) graphs, find upper and lower bounds on the (planar) slope number of any graph G in the family The (Planar) Slope Number Problem

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Given a family of (planar) graphs, find upper and lower bounds on the (planar) slope number of any graph G in the family The (Planar) Slope Number Problem

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Motivation and State of the Art

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Planar bounded-degree graphs have bounded planar slope number - psl(G) O(2 O( ) - [Keszegh et al., SIAM J. Discrete Math., 2013]

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Motivation and State of the Art Is the slope number of “nearly-planar” graphs bounded in ? Planar bounded-degree graphs have bounded planar slope number - psl(G) O(2 O( ) - [Keszegh et al., SIAM J. Discrete Math., 2013]

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Motivation and State of the Art The planar s. n. of planar partial 3-trees is O( ) [Jelínek et al.,Graphs & Combinatorics, 2013]

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Motivation and State of the Art The planar s. n. of planar partial 3-trees is O( ) [Jelínek et al.,Graphs & Combinatorics, 2013] The planar s. n. of partial 2-trees is ( ) [Lenhart et al., GD 2013]

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Motivation and State of the Art The planar s. n. of planar partial 3-trees is O( ) [Jelínek et al.,Graphs & Combinatorics, 2013] The planar s. n. of partial 2-trees is ( ) [Lenhart et al., GD 2013] Can this gap be reduced for subfamilies of planar partial 3-trees?

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Our research outer 1-planar graphs: graphs that admit drawings where each edge is crossed at most once and each vertex is on the boundary of the outer face

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Our research there exists a linear-time algorithm for testing outer 1-planarity (returns an o1p embedding) [Auer et al., GD 2013] [Hong et al., GD 2013] K4K4 o1p drawing

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Our research there exists a linear-time algorithm for testing outer 1-planarity (returns an o1p embedding) o1p graphs are planar partial 3-trees [Auer et al., GD 2013] [Hong et al., GD 2013] K4K4 o1p drawingplanar drawing

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Results The o1p slope number, o1p-sl(G), of o1p graphs with maximum degree Δ is at most 6Δ This result also generalizes the result on the planar s. n. of outerplanar graphs which is Δ – 1 [Knauer et al., CGTA, 2014]

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Results The planar slope number of o1p graphs with maximum degree Δ is O(Δ 2 ) The o1p slope number, o1p-sl(G), of o1p graphs with maximum degree Δ is at most 6Δ This result also generalizes the result on the planar s. n. of outerplanar graphs which is Δ – 1 [Knauer et al., CGTA, 2014]

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Results The planar slope number of o1p graphs with maximum degree Δ is O(Δ 2 ) The o1p slope number, o1p-sl(G), of o1p graphs with maximum degree Δ is at most 6Δ This result also generalizes the result on the planar s. n. of outerplanar graphs which is Δ – 1 [Knauer et al., CGTA, 2014]

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Overview of proof technique

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Prove o1p-sl(G) ≤ Δ for 2-connected o1p graphs

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Overview of proof technique Prove o1p-sl(G) ≤ Δ for 2-connected o1p graphs Extend the result to 1-connected o1p graphs

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Overview of proof technique Prove o1p-sl(G) ≤ Δ for 2-connected o1p graphs Extend the result to 1-connected o1p graphs

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Overview of proof technique Prove o1p-sl(G) ≤ Δ for 2-connected o1p graphs Ingredients: a universal set of slopes bottom-up visit of SPQR-tree

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A universal set of slopes

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b1b1

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b1b1 r + 1 r - 1

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A universal set of slopes In the following: edges using blue (red) slopes will never (always) receive crossings b1b1 r + 1 r - 1

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SPQR-tree decomposition G is a 2-connected outer 1-plane graph G Recursive decomposition based on the split pairs into its triconnected components Four types of nodes: S,P,Q,R

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SPQR-tree decomposition P S S P Q Q Q Q R G P Q T S Q Q S Q Q S Q Q Q QQ QQ

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sµsµ tµtµ G G µ = pertinent graph of µ the pertinent graph G µ of µ is the subgraph of G whose SPQR-tree is the subtree of T rooted at µ P S S P Q Q Q Q R P Q T µ S Q Q S Q Q S Q Q Q QQ QQ s µ, t µ = poles of µ

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SPQR-tree decomposition sµsµ tµtµ G The skeleton of µ is the biconnected (multi)graph obtained from G µ by collapsing the pertinent graphs of the children of µ with a virtual edge, plus the reference edge P S S P Q Q Q Q R P Q T µ S Q Q S Q Q S Q Q Q QQ QQ

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SPQR-tree decomposition P S S P Q Q Q Q R G P Q T S Q Q S Q Q S Q Q Q QQ QQ key property 1: internal P-nodes have at most three children and, in this case, two are S-nodes that cross and the other one is either a Q-node or an S-node

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SPQR-tree decomposition P S S P Q Q Q Q R G P Q T S Q Q S Q Q S Q Q Q QQ QQ key property 2: the skeleton of an R-node is isomorphic to K 4 and embedded with one crossing

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SPQR-tree decomposition P S S P Q Q Q Q R G P Q T S Q Q S Q Q S Q Q Q QQ QQ key property 3: two Q-nodes at different levels of T cross only if one of them is the child of an S-node and the other one the grandchild of a P-node that is also a child of

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SPQR-tree decomposition: Postprocessing P S S S* Q Q Q R G P Q T S Q Q S Q Q S Q Q Q QQ QQ S*-node = merge of a P-node + a Q-node that are children of the same S-node and that ‘cross’ each other Q

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2-connected o1p graphs: Proof scheme Let T be an SPQR-tree of an embedded o1p graph G

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2-connected o1p graphs: Proof scheme Visit the nodes of T in a bottom-up order: for each node µ of T draw G µ by combining the drawings of the children of µ Let T be an SPQR-tree of an embedded o1p graph G

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Invariants: I1. Γ µ is o1p I2. Γ µ uses only slopes of the universal slope set I3. 2-connected o1p graphs: Proof scheme Visit the nodes of T in a bottom-up order: for each node µ of T draw G µ by combining the drawings of the children of µ Let T be an SPQR-tree of an embedded o1p graph G sµsµ tµtµ ΓµΓµ β µ < [Δ(s µ )+1]α γ µ < [Δ(t µ )+1]α βµβµ γµγµ

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How to draw S-nodes

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sµsµ tµtµ S P P … R We attach the (already computed) drawings of the pertinent graphs of the children of the current node sµsµ tµtµ

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How to draw S-nodes sµsµ tµtµ S P … R The resulting drawing is o1p (I1.) and uses only slopes in the universal slope set (I2.) P

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How to draw S-nodes sµsµ tµtµ S P … R The drawing is contained in a triangle that respects the third invariant (I3.) βµβµ γµγµ β µ < [Δ(s µ )+1] α γ µ < [Δ(t µ )+1] α P

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How to draw P-nodes

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P SS sµsµ tµtµ Q QQ … …

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P SS sµsµ tµtµ Q QQ … …

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b2Δb2Δ b2b2 P SS sµsµ tµtµ Q QQ … …

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r - 2 r + 2Δ b2Δb2Δ b2b2 P SS sµsµ tµtµ Q QQ … …

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How to draw P-nodes sµsµ tµtµ b2b2 b2Δb2Δ r - 2 r + 2Δ P SSQ QQ … … r - 2 r + 2Δ b2Δb2Δ b2b2

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How to draw P-nodes sµsµ tµtµ b2b2 b2Δb2Δ r - 2 r + 2Δ b1b1 P SSQ QQ … …

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How to draw P-nodes sµsµ tµtµ b2b2 b2Δb2Δ r - 2 r + 2Δ The drawing is o1p (I1.) and uses only slopes in the universal slope set (I2.) b1b1 P SSQ QQ … …

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How to draw P-nodes sµsµ tµtµ b2b2 b2Δb2Δ r - 2 r + 2Δ The drawing is contained in a triangle that respects the third invariant (I3.) β µ < [Δ*(s µ )+1] α + α < [Δ(s µ )+1] α γ µ < [Δ*(t µ )+1] α + α < [Δ(t µ )+1] α * * b1b1 P SSQ QQ … …

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How to draw S*-nodes

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S* S SS sµsµ tµtµ Q

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How to draw S*-nodes sµsµ tµtµ sµsµ b2Δb2Δ r - 2 r + 2Δ The drawing is o1p (I1.) and uses only slopes in the universal slope set (I2.) tµtµ S* S SS Q

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How to draw S*-nodes sµsµ b2Δb2Δ r - 2 r + 2Δ Is the drawing contained in a triangle that respects the third invariant (I3.)? tµtµ S* S SS Q

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How to draw S*-nodes sµsµ b2Δb2Δ r - 2 r + 2Δ We need the line with slope b 3 passing through s µ not to intersect τ*: the height of τ* depends on ε b3b3 β µ < [Δ(s µ )+1] α = 2α = b 3 tµtµ βµβµ τ*τ* S* S SS Q

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How to draw S*-nodes τ*τ* 0 < ε < α

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How to draw R-nodes

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sµsµ r - 2 r + 2Δ tµtµ Similar construction as for S*-nodes

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Open problems

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Study the 1-planar s. n. of 1-planar straight-line drawable graphs: Is it bounded in Δ?

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Open problems Is the quadratic upper bound for planar s.n. of o1p graphs tight? Study the 1-planar s. n. of 1-planar straight-line drawable graphs: Is it bounded in Δ?

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Open problems Can we draw planar partial 3-trees with o(Δ 5 ) slopes and a constant number of crossings per edge? Is the quadratic upper bound for planar s.n. of o1p graphs tight? Study the 1-planar s. n. of 1-planar straight-line drawable graphs: Is it bounded in Δ?

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Open problems Can we draw planar partial 3-trees with o(Δ 5 ) slopes and a constant number of crossings per edge? Is the quadratic upper bound for planar s.n. of o1p graphs tight? Study the 1-planar s. n. of 1-planar straight-line drawable graphs: Is it bounded in Δ? Thank you!

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