introduction to graph drawing - university...

INTRODUCTION TOGRAPH DRAWINGFall 2010

Battista G D Eades P Tamassia R and Tollis I G 1998 Battista, G. D., Eades, P., Tamassia, R., and Tollis, I. G. 1998 Graph Drawing: Algorithms for the Visualization of Graphs. 1st. Prentice Hall PTR.

PLANARITY TESTING

PLANARITY TESTING

1. Count edges and check Euler's formulag2. Find pieces of G3. For each piece P that is not a path

test planarity by recursion4. Compute interlacement graph of the pieces5 T t if th i t l t h i bi tit5. Test if the interlacement graph is bipartite

PLANARITY TESTING

Graph is planar if and only if all its connected p p ycomponents are planarA connected graph is planar if and only if all its bi t d t lbiconnected components are planar

REVIEW

A graph is connected if there is a path between ug p pand v for each pair (u,v) of vertices A cutvertex in a graph is a vertex whose removal di t th hdisconnects the graph.A connected graph with no cutvertices is biconnected.biconnected.A maximal biconnected subgraph of a graph is biconnected component.

Decomposing the Graph into connected and p g pbiconnected components. Our problem will be restricted to testing the

l it f bi t d hplanarity of biconnected graphs.Use cycle to decompose a biconnected graph into pieces.pieces.

PIECE P OF A GRAPH G WITH RESPECT TOPATH C

The subgrap induced by the edges of path C in a g p y g pclass is called a piece of G with respect to C

v6v4

v7v2

v7v2v3

4 v9

v1v5

' ( ' ')G V Ev9

v1v5

v8

1

v2

' ( ', ')'' ( ' ')

G V EV VE E V V

=⊆⊆ ∩ ×

( , )G V E= v8

1 v7

v9

v2

' ( ', ')'' ( ' ')

G V EV VE E V V

=⊆= ∩ × v8

v1v5

PARTITIONING INTO PIECES

Let a biconnected graph G contain a cycle Cg p yPartition the edges of G not on C into classes:

Two edges of G are in the same class if there is a path b t th th t d t t i t f Cbetween them that dose not contain any vertex of C.

SUB GRAPH INDUCED BY EDGES IN A CLASS ISCALLED A PIECE OF G WITH RESPECT TO C.

Pieces consisting of a single edge between two Pieces consisting of a single edge between two vertices of CPieces consisting of a connected graph with at least one vertex not in C

ATTACHMENTS

Vertices of piece P which are also on cycle C are p ycalled attachments of P.Each piece has at least two attachments. (why?)

SEPARATING CYCLE

Cycle C is separating if it has at least two pieces.y p g pAnd it is nonseparating if it has one piece.

separatingnonseparating

LEMMA 3.4Let G be a biconnected graph and C be a g pnonseparating cycle of G with piece P. If P is not a path then G has a separating cycle C’ consisting of subpath of C and a path of Pof subpath of C and a path of P.

separatingnonseparating

LEMMA 3.4 PROOF

Let u and v be 2 attachments of P that areti i th i l d iconsecutive in the circular ordering.

Let γ be a subpath of C between u and vwithout any attachments.Si P i t d th i γ

π

Since P is connected there is apath π in P between u and v.Let C’ be the cycle obtained from Cby replacing γ with π C

γe

by replacing γ with π.Now γ is a piece on G with respect to C’Let e be an edge on P not π.e exist Because P is not a pathe exist Because P is not a path.So there is a piece of C’ otherthan γ which contains e.Thus C’ is a separating cycle in G

C’

Thus C is a separating cycle in G.

INTERLACEMENT

Each piece can be drawn either entirelyp yinside or outside of the cycleInterlacing pieces are ones that can’t be drawn on th id f C ith t ithe same side of C without crossing

Interlace

Not interlace

INTERLACEMENT GRAPH

Vertices are pieces on G with respect to cycle Cp p yEdges are pairs of pieces that interlace (can’t reside on the same side of C without crossing).

INTERLACEMENT TO PLANARITY

If G is planar graph then its interlacement graph p g p g pmust be bipartite

P3P1

P5P6

P2P4

P2

THEOREM 3.8A biconnected graph G with cycle C is planar g p y pif and only if:For each piece P of G with respect to C, the graph bt i d b ddi P t C i lobtained by adding P to C is planar

The interlacement graph of the pieces of G with respect to C, is bipartiterespect to C, is bipartite

PLANARITY TESTING

1. Count edges and check Euler's formulag2. Find pieces of G3. For each piece P that is not a path

test planarity by recursion4. Compute interlacement graph of the pieces5 T t if th i t l t h i bi tit5. Test if the interlacement graph is bipartite

ALGORITHM PLANARITY TESTING

Input a biconnected graph G with n vertices and at p g pmost 3n-6 edges, and a separating cycle C.

Output an indication of whether G is planar

1 C t th i f G ith t t C (O( ))1. Compute the pieces of G with respect to C. (O(n))

ALGORITHM PLANARITY TESTING

2. For each piece P of G that is not a path:p pLet P’ be the graph obtained by adding P to CLet C’ be the cycle of P’ obtained from C by replacing the portion of C between two consecutive attachments the portion of C between two consecutive attachments with a path P between themApply the algorithm recursively to graph P’ and cycle C’ If P’ i l t “ l ” (O( 2)) C’. If P’ is nonplanar, return “nonplanar”. (O(n2))

ALGORITHM PLANARITY TESTING

3. Compute the interlacement graph I of the p g ppieces. (O(n2))

ALGORITHM PLANARITY TESTING

4. Test whether I is bipartite. If I is not bipartite p preturn “nonplanar”. O(n2))

5. Return “planar”.

OVERALL RUNTIME

Each recursive invocation takes O(n2)( )Each recursion is associated with at least one edge of GThere are O(n) edgesRuntime is O(n3)