p lanarity t esting a nd e mbedding by: seyed akbar mostafavi fall 2010 1
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PLANARITY TESTING AND EMBEDDING
By: Seyed Akbar Mostafavi
Fall 2010
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OUTLINE
Planar Graphs Planarity Testing Theorems
Kuratowski’s Theorem Demoucron’s Planarity Algorithms Euler Polyhedron Formula Nonplanarity of Kuratowski Graphs Minimum Degree Bound
Definitions of Planarity Testing st-numbering Algorithm Bush Form and PQ-Tree Booth-Lueker Algorithm Embedding Algorithms
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PLANAR GRAPHS
A graph G(V,E) is called planar graph if it can be drawn on embedded in the plane in such a way that the edges of the embedding intersect only at the vertices of G.
Example: K(4)
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PLANAR GRAPHS
Nonplanar Graph example: K(3,3)
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PLANARITY TESTING
Two classical approaches Theorem of Kuratowski
Characterizes planar graphs in terms of subgraphs they are forbidden to have
Based on the notion of homeomorphism Series insertion: replacement of an edge (u,v) of G by
a pair of edges (u,z) and (z,v) Series deletion: replacement of a pair of edges (u,z)
and (z,v) of G by a single edge (u,v) Hopcraft and Tarjan Algorithm
determines planarity in linear time and shows how to draw the graph if it is planar
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KURATOWSKI’S THEOREM
A graph G(V,E) is planar if and only if G contains no subgraph homeomorphic to either K(5) or K(3,3)
Example: Petersen’s Graph
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CONTRACTION FORM OF KURATOWSKI’S THEOREM
Contraction: operation of removing an edge (u,v) from a graph G and identifying the endpoints u and v (with a single new vertex uv) so that every edge originally incident with either u and v becomes incident with uv.
Graph (V,E) is said contractible to a graph (V,E) if can be obtained from by a sequence of contractions.
Theorem: A graph G(V,E) is planar if and only if G contains no subgraph contractible to either K(5) or K(3,3).
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DEMOUCRON’S PLANARITY ALGORITHM Terminology
face of a planar representation: a planar region bounded by edges and vertices of the representation and containing no graphical elements in its interior.
R: a planar representation of a subgraph S of a graph G(V,E)
We try to extend R to a planar representation of G Constraints
Each component of G-R, being a connected piece of the graph, must, in any planar extension of R, be placed within a face of R.
Further constraint is observed by considering those edges connecting a component c of G-R to set of vertices W in R.
All the vertices W in R must lie on the boundary of a single face in R. Otherwise, c would have to straddle more than one face of R.
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TERMINOLOGY (MORE)
A part p of G relative to R is defined as1) An edge (u,v) in E(G) – E(R) such that u and v or2) A component of G – R together with the edges
(called pending edges) that connect the component to the vertices of R.
Contact vertex of a part p of G relative to R is defined as a vertex of G – R that is incident with a pending edge of p
A planar representation R of a subgraph S of a planar graph G is planar extensible to G if R can be extended to a planar representation of G.
A part p of G relative to R is drawable in a face of R if there is a planar extension of R to G where p lies in f.
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DEMOUCRON’S ALGORITHM Idea: we repeatedly extend a planar representation
R of a subgraph of graph G until either R equals G or the procedure becomes blocked in which case G is nonplanar.
1) We start by taking R to be an arbitrary cycle from G
2) We then partition the remainder of G not yet included in R into parts of G relative to R and apply drawability condition.
3) We identify those faces of R in which p is potentially drawable, a path q through p, and add q to R by drawing it through f in a planar matter.
4) Process is blocked only if we find some part that doesn’t satisfy the drawability condition for any face in which case R is not planar extensible to G.
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DECOUCRON’S ALGORITHM PSEUDO-CODEFunction Draw (G, R)(* Returns a planar representation of G in R)Var G: Graph
R: Planar Representationp: PartP: set of Partf: FaceF: Set of FaceDraw: Boolean function
Set Draw to TrueSet R to some cycle in G
repeatSet P to the set of parts of R relative to Gfor each p in P do Set F(p) to the set of faces of R
in which p is drawableif F(p) is empty for any pthen Set Draw to False
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DECOUCRON’S ALGORITHM PSEUDO-CODE
else if For some part p, F(p) contains only a single facethen Let f be the face in which p is drawableelse Let p be any part and let f be a face in F(p)
Let q be a path between a pair of contact vetices of p with R that contains only edges of pSet R to
until R = G or not Draw
End_function_Draw
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EXAMPLE OF CONSTRUCTION OF PATH
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PLANAR GRAPH THEORY
Theorem (Euler Polyhedron Formula). Let G(V,E) be a connected planar graph and |F| denote the number of faces in a planar embedding of G. Then
Proof: induction on the number of edge
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PLANAR GRAPH THEORY (CONT’) Theorem (Linear bound on |E|). If G(V, E) is a
planar graph, then
Proof: We let denote the number of edges bounding the face of some planar representation of G. Each face must contains at least 3 bounding edges. Therefore,
Since each edge borders exactly two faces, the summation counts each edges twice. Therefore,
Combining these results and using the Euler Polyhedron Formula, we obtain
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PLANAR GRAPH THEORY (CONT’)
Theorem (Nonplanarity of Kuratowski Graphs). The graphs K(3,3) and K(5) are nonplanar.
Proof. Nonplanarity of K(5) follows by contradiction from the Linear Bound Theorem. For K(3,3), we argue as follows.
By Euler’s Theorem, |F| = |E| - |V| + 2 = 5. However, since every cycle in K(3,3) has at least four edges, similar to proof of Linear Bound Theorem, we would have
whence emulating the Linear Bound proof, we could conclude that so that , contrary to the previous lower bound of 5.
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PLANAR GRAPH THEORY (CONT’)
Minimum Degree Bound. If G(V,E) is a planar graph, .
Proof (by contradiction). Suppose that the minimum degree of G is at least 6. then
Since the sum of the degrees is always 2|E|, it follows that |E| is at least 3|V|, contrary to the Linear Bound Theorem.
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VERTEX ADDITION ALGORITHM
Vertex Addition Algorithm is based on s-t numbering, that assigns an index to each vertex.
Graph is represented by a set of n lists, called “adjacency lists” Adj(v)
The adjacency list of a vertex contains all its neighbors
An embedding of a graph determines the order of the neighbors embedded around a vertex.
A graph is planar if and only if all the 2-connected components are planar [Har72].
Corollary 1.1: ; otherwise the graph is nonplanar
From: Planar Graphs
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DEFINITION OF S-T NUMBERING
An st-numbering is numbering 1, . . . , n of the vertices of a graph such that Vertices “1” and “n” are adjacent Every other vertex j is adjacent to two vertices i
and k such that Vertex “1” is the source s and vertex “n” the
sink t Every 2-connected graph G has an st-
numbering, and an algorithm given by Even and Tarjan finds an st-numbering in linear time.
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DEPTH-FIRST SEARCH
Start with arbitrary edge (t, s) Compute for each vertex its depth-first
number DFN(v), its parent FATH(v) and its lowpoint LOW(v)
LOW(v) = min({v} {w| there exists a back edge (u,w) such that
u is a descendant of v and w is an ancestor of v in a DFS tree T})
Definition( Lowpoint)
LOW(v) = min({v} {LOW(x)|x is a son of v} {w|(v,w) is a back edge}
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DFS ALGORITHMProcedure DFS(G);
procedure SEARCH(v)
begin
mark v “old”;
DFN(v) := COUNT;
COUNT := COUNT + 1;
LOW(v) := DNF(v);for each vertex w on Adj(v)
do if w is marked “new”
then begin
add (v,w) to T
FATH(w):=v;
SEARCH(w);
LOW(v) := min{LOW(v) , LOW(w)}
end
else if w is not FATH(v)
then LOW(v) := min {LOW(v), DFN(w)}
end;
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PARTITION EDGES TO PATHS Vertices s, t and edge (s, t) are marked “old”Case 1 There is a “new” back edge (v,w)
Mark (v,w) “old” Return vw
Case 2 There is a “new” tree edge (v,w) Let w w1 w2 . . .wk be the path to the lowpoint wk of v Mark vertices and edges on the path “old” Return v w w1 w2 . . .wk
Case 3 There is a “new” back edge (w, v) Let w w1 w2 . . .wk be the path going backward to an old
vertex wk Mark vertices and edges on the path “old” Return v w w1 w2 . . .wk
Case 4 All edges incident to v are “old” Return
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ST-NUMBERING
Special nodes: source s (“1”) and sink t (“n”)“1” and “n” adjacent j V-{s,t}: i, k Adj(j)st(i)<st(j)<st(k)
So 2 must be next to 1and n-1 must be next to n
4 6
2
5
71
3
8
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SOME ST-NUMBERING EXAMPLES
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1
11
2
10
3
9
4
8
5
7
6
31
5
2
4
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ST-NUMBERING
Theorem:Every biconnected graph has an st-numbering
Proof:We give an algorithm and show that it works if the graph is biconnected
Our graph is biconnected, so anst-numbering exists.
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FINDING AN ST-NUMBERING
First step: standard DFS. Can visit children in any order Calculate and store per vertex:
Parent “Depth first search number:” DFN Low-point: the lowest DFN among nodes that can be
reached by a path using any amount of tree edges followed by 1 back edge
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G
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DFS
G
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1
2
3
4
5
6 DFN
DFS
G
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1
2
3
4
5
6DFN
1
1
1
1
3
3LOW
DFS
G
30
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FINDING AN ST-NUMBERING
Push the vertices onto a stack and pop them off Four rules for what to push and pop Assign increasing numbers to the vertices when
they are popped off for the last time
Example follows
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INITIAL SETUP
The node with DFN 2 is called the ‘source’ The node with DFN 1 is called the ‘sink’
Stack with sourceon top of sink
Source and sinkare “old”
c
f
d
b
e
a
ac
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RULE “2”
There is a “new” tree edge: follow it and take the path that leads to its low point
Push the vertices on the path ontothe stack
Mark all as old.
c
f
d
b
e
a
abc
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NO MATCHES FOR VERTEX A
There are now no rule matches for a. Pop it off the stack and give it the next
available number This node is ready
c
f
d
b
e
1
bc
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RULE “3”
There is a “new” back edge into this node: from the other end, go back up tree edges to an “old” vertex
Push the vertices onthe path ontothe stack
Mark all as old
c
f
d
b
e
1 bfedc
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NO MATCHES FOR VERTEX B
So the vertex is ready and gets its number
c
f
d
2
e
1fedc
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NO MATCHES FOR VERTEX F
So the vertex is ready and gets its number
c
3
d
2
e
1
edc
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ET CETERA…
6
3
5
2
4
1
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EXAMPLE OF S-T NUMBERING
1
2
3
4
5
6
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BUSH FORM
Let = ( , ) be the subgraph induced by the vertices = {1, . . . , k}
Let be the graph formed by adding all edges with ends in V − , where the ends of the edges are kept separate
These edges are called virtual edges and their ends virtual vertices
A bush form of is an embedding of such that the virtual vertices are on the outer face
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EXAMPLE OF BUSH FORM
st-numbering
𝐺4 bush form
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PQ-TREE DATA STRUCTURE
Use PQ-tree to represent bush form PQ-tree consists of
P-nodes Represents a cut vertex of , and its children can be permuted arbitrarilyQ-nodes Represents a 2-connected component of , and its children are only allowed to reverseleaves Represents a virtual vertex of
PQ-tree represents all the permutations and reversions possible in a bush form
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EXAMPLE OF PQ-TREE
bush form PQ-tree
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VERTEX ADDITION ALGORITHM
Idea of the algorithm is to test planarity of +1 by finding these permutations and reversions
The permutations and reversions can be found by applying 9 transformation rules to the PQ-tree (template matching)
A leaf labeled “k + 1” is pertinent and a pertinent subtree is a minimal subtree of a PQ-tree containing all the pertinent leaves
A node of a PQ-tree is full if all the leaves of its descendants are pertinent
If we have a bush form of a subgraph of a planar graph G, then there exists a sequence of permutations and reversions to make all virtual vertices labeled “k + 1” occupy consecutive positions
Lemma
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TEMPLATE MATCHINGS
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TEMPLATE MATCHINGS
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TEMPLATE MATCHINGS
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TEMPLATE MATCHINGS
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PQ-TREE REDUCTION EXAMPLE
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PLANARITY TESTING ALGORITHM
1. Assign st-numbers to the vertices of G
2. Construct PQ-tree corresponding to 3. Gather pertinent leaves by applying
the template matching4. If the reduction fails then G is
nonplanar5. Replace full nodes of the PQ-tree by
a new P-node6. Goto 3
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM
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EXAMPLE OF VERTEX ADDITION ALGORITHM