Flow and Upward Planarity

Akil Kömür

Introduction

• Upward planar: admits drawing that’s upward and planar

• Digraphs• 1) characterization that relates upward planarity

with st-graphs• 2) angles which play a big role in upward

planarity• 3) polynomial time algorithm to test upward

planarity of embedded acyclic digraphs

Introduction

• Two acyclic drawings

1) Inclusion in a Planar st-Graph

• Planar st-graph is an st-graph that is planar and embedded with vertices s and t on the boundary of the external face

• St-graph is an acyclic graph with a single source s and single sink t

• Simple characterization of upward planarity:

1) Inclusion in a Planar st-Graph

• Example theorem 6.1:

1) Inclusion in a Planar st-Graph

• Statement 1 implies Statement 3• S,T respc. lowest and highest y coord.• At sink v ≠ t draw new edge upward• Add new edges (v, dest(e))• Able to cancel all sinks, except t• Similar process cancels all sources, except s• Finally edge (s,t) added• Statement shown.

1) Inclusion in a Planar st-Graph

• Statement 3 implies Statement 2• Given planar st-graph G′ including G construct a

planar straight-line drawing in 3 steps• 1-add edges to G′. Resulting digraph G′′ is

planar st-graph with all faces consisting of three edges

• 2-construct upward planar straight-line drawing of G′′

• 3-remove edges that don’t belong to G from drawing of G′′

1) Inclusion in a Planar st-Graph

• Step 1 in figure• Adding edges until all faces have three edges.

1) Inclusion in a Planar st-Graph

• Step 2 - construct upward planar straight-line drawing of G′′

• The kernel of X is the set of points p from which all vertices of X are visible.

• The wedge of vertex vi of X is defined as the intersection

of the left half planes

1) Inclusion in a Planar st-Graph

• Lemma 6.3: used in proof Lemma 6.4

1) Inclusion in a Planar st-Graph

• Proof: omitted

• Implies that statement 2 of theorem 6.1 implies statement 3

1) Inclusion in a Planar st-Graph

• Statement 3 implies statement 1 (easy)• But uses algorithm discussed in chapter 4

(Battista)• Lemma 6.5 yields O(n) algorithm for constructing

an upward planar straight-line drawing of an n-vertex planar st-graph

• Testing if a digraph is planar st-Graph in O(n)• G has single source and single sink• G plus the edge (s,t) is planar (as an undirected graph)• G is acyclic -> tested in O(n)

1) Inclusion in a Planar st-Graph

• Theorem 6.2:– Upward planarity testing is in NP.– Proof in paper.

2) Angles in Upward Drawings

• Characterization of upward planarity for embedded digraphs. Algorithm for this follows in 3)

• Vertex is bimodal if cyclic sequence of its incident edges can be partitioned into two (possibly empty) linear sequences, one for incoming the other for outgoing

• Lemma 6.5– An embedded digraph is upward planar only if it is

bimodal.

• Planar st-graph is bimodal, follows from L6.5, and fact that planar st-graph is upward planar.

2) Angles in Upward Drawings

• Assignments of labels {small, large} on pairs of incoming/ outgoing edges (straight line drawing)

• p is either face r vertex• L(p): nr of angles of p with label large• S(p): nr of angles of p with label small• A(f): the number of angles in f formed by pairs of

incoming (or outgoing) edges• L(f) + S(f) = 2A(f)

– See example

2) Angles in Upward Drawings

2) Angles in Upward Drawings

• Motivated by above formulation: Assignment that maps each vertex v to a face incident to v.

2) Angles in Upward Drawings

2) Angles in Upward Drawings

• Lemma 6.8 deduced from Lemma 6.7– An embedded bimodal digraph is upward planar only

if it admits a consistent assignment of sources and sinks to faces.

• Now we can use Algorithm 6.1 to construct a planar st-graph that includes the embedded bimodal digraph as a sub graph

2) Angles in Upward Drawings

• Assigns labels sL tL and sS tS to each face in a circular sequence of symbols by traversing the face clockwise.

2) Angles in Upward Drawings

• Example of behavior of the algorithm

2) Angles in Upward Drawings

• Proof omitted. Must show that each edge insertion preserves planarity, acyclicity, and bimodality. And have exactly one source and sink.

• Algorithm 6.2 linear in nr of vertices of f• Algorithm 6.1 takes O(n)-time• Summarize this in Theorem 6.3

3) Upward Planarity Testing

• Algorithm for testing whether an embedded digraph G is upward planar. Based on theorem 6.3.

• To test if G admits consistent assignment of s and t: construct a bipartite flow network Bh

• Example figure follows.

• Lemma 6.10 summarizes the properties of Bh

3) Upward Planarity Testing

3) Upward Planarity Testing

• Constructing Bh takes O(n) time

• Existence of flow tested in O(rn) time• So total is O(n2r)• Time complexity can be reduced to O(nr) by using

Algorithm 6.3

3) Upward Planarity Testing

• Algorithm 6.3:• Construct a flow network B, like a Bh but demand

is A(f) – 1. O(n)• Test whether B admits a flow of value r-2. O(n-2)• For each face the demand is increased by two

units and a test is done to see whether the flow of value r-2 in B can be augmented by two units O(nr)

• The set of all faces of G for which step 3 was successful is returned.

3) Upward Planarity Testing

• Testing whether embedded digraph G is acyclic and bimodal takes O(n) time

• Then using Algorithm 6.3 to test whether G admits a consistent assignment of sources and sinks to its faces takes O(nr) time

• Lemma 6.4– Let G be an embedded digraph with n vertices and r

sources and sinks. We test whether G is upward planar in O(nr) = O(n2) time

Summary

• 1) characterization that relates upward planarity with st-graphs

• 2) angles which play a big role in upward planarity

• 3) polynomial time algorithm to test upward planarity of embedded acyclic digraphs

Questions?