Convex drawingConvex drawingchapter 5chapter 5

Ingeborg GroenewegIngeborg Groeneweg

SummerySummery What is convex drawingWhat is convex drawing Some definitionsSome definitions Testing convexityTesting convexity Drawing a convex graphDrawing a convex graph

Convex drawingConvex drawing Drawing is called convex:Drawing is called convex:

Each egde straight lineEach egde straight line Each face convex polygonEach face convex polygon

Not every planar graph is convexNot every planar graph is convex Every 3-connected planar graph has Every 3-connected planar graph has

a convex drawinga convex drawing

Facial cycleFacial cycle Boundary of a face Boundary of a face Facial cycle of a graph G is boundary Facial cycle of a graph G is boundary

of outer faceof outer face Facial cycle of G also called CFacial cycle of G also called C00(G)(G) C*C*00, outer convex polygon, polygonal , outer convex polygon, polygonal

drawing of Cdrawing of C00

ExtendibleExtendible C*C*00 is extendible if there exists a is extendible if there exists a

convex drawing of G with Cconvex drawing of G with C00(G) drawn (G) drawn as C*as C*00

Let C*Let C*00 be a k-gon, k be a k-gon, k ≥ 3≥ 3 PP11, P, P22, .. ,P, .. ,Pkk paths in C paths in C00(G), (G),

corresponding to a side of C*corresponding to a side of C*00

C*C*00 is extendible if and only if is extendible if and only if Condition I holdsCondition I holds

Condition ICondition I For each inner vertex v with d(v) For each inner vertex v with d(v) ≥≥ 3, 3,

there exists three paths disjoint except there exists three paths disjoint except v, each joining v and an outer vertexv, each joining v and an outer vertex

G – V(CG – V(Coo(G)) has no connected (G)) has no connected component H such that all the outer component H such that all the outer vertices adjacent to vertices in H lie on vertices adjacent to vertices in H lie on a single path Pa single path Pii are joined by an inner are joined by an inner edgeedge

Any cycle containing no outer edge has Any cycle containing no outer edge has at least three vertices of degree at least three vertices of degree ≥ ≥ 33

DefinitionsDefinitions Separation pairSeparation pair Split componentSplit component 3-connected component3-connected component Prime separation pairPrime separation pair Forbidden separation pairForbidden separation pair Critical separation pairCritical separation pair

Separation pairSeparation pair Two subgraphs GTwo subgraphs G11 = (V = (V11, E, E11) , G) , G22 = =

(V(V22,E,E22) of 2-connected graph G = ) of 2-connected graph G = (V,E)(V,E)

(x,y) (x,y) V is separation pair if V is separation pair if V = V1 V = V1 V2 , {x,y}= V1 V2 , {x,y}= V1 V2 V2 E = E1 E = E1 E2 , E2 , = E1 = E1 E2 E2 E1E1 ≥ 2 , ≥ 2 , E2E2 ≥ 2 ≥ 2

Separation pairSeparation pair exampleexample

Split graphsSplit graphs Split graphs: obtained by adding a virtual Split graphs: obtained by adding a virtual

edge (x,y) to Gedge (x,y) to G11 and G and G22

Splitting: dividing graph into two split Splitting: dividing graph into two split graphsgraphs

Split components: splitting (split) graphs Split components: splitting (split) graphs until no more splits are possibleuntil no more splits are possible

3-connected components3-connected components merging split componentsmerging split components

Triple bonds into a bondTriple bonds into a bond Triangles into a ring(= a cycle)Triangles into a ring(= a cycle)

3-connected components are unique3-connected components are unique1

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Prime separation pairPrime separation pair Prime separation pair {Prime separation pair {x,yx,y}:}:

xx and and y y end vertices of virtual edge end vertices of virtual edge contained in 3-connected componentcontained in 3-connected component

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Forbidden separation Forbidden separation pairpair

Prime separation pair is forbidden if:Prime separation pair is forbidden if: At least four {x,y}-split components, orAt least four {x,y}-split components, or Exactly three {x,y}-split components: Exactly three {x,y}-split components:

no ring, no bondno ring, no bond

Critical separation pairCritical separation pair Prime separation pair is critical if:Prime separation pair is critical if:

Exactly three {x,y}-split components Exactly three {x,y}-split components including a ring or a bond, orincluding a ring or a bond, or

Exactly two {x,y}-split components:Exactly two {x,y}-split components:no ring, no bondno ring, no bond

=

=

Condition IICondition II Let C*Let C*0 0 be outer strict convex be outer strict convex

polygon polygon G has no forbidden separation pairG has no forbidden separation pair For each critical separation pair (x,y) of For each critical separation pair (x,y) of

G, there is at most one (x,y)-split G, there is at most one (x,y)-split component having no edge of F, and if component having no edge of F, and if any, it is either a bond if (x,y) any, it is either a bond if (x,y) E or a E or a ring otherwisering otherwise

Testing convexityTesting convexity Forbidden separtion pair Forbidden separtion pair no convex no convex

drawingdrawing No forbidden, one critical No forbidden, one critical convex convex

drawing some outer facial cycledrawing some outer facial cycle No forbidden, two or more critical No forbidden, two or more critical

further specificationfurther specification No forbidden, No critical No forbidden, No critical convex convex

drawing for any facial cycle, drawing for any facial cycle, subdivision of 3-connected graphsubdivision of 3-connected graph

Testing convexityTesting convexity For every critical separation pair (x,y)For every critical separation pair (x,y)

(x,y) (x,y) E -> delete (x,y) E -> delete (x,y) (x,y) (x,y) E and one (x,y)-split component is E and one (x,y)-split component is

ring -> delete x-y pathring -> delete x-y path Resulting graph G’Resulting graph G’ Add to G’ new vertex vAdd to G’ new vertex v Join v to all critical separation verticesJoin v to all critical separation vertices If new Graph G’’ is planar <-> G has If new Graph G’’ is planar <-> G has

convex drawingconvex drawing

Finding convex drawingFinding convex drawing Find a extendible facial cycle FFind a extendible facial cycle F Remove all vertices v, with d(v)=2 and v Remove all vertices v, with d(v)=2 and v F F Remove w Remove w F + all edges incident to w F + all edges incident to w Devide G’= G – w in blocksDevide G’= G – w in blocks Determine outer convex polygon for each blockDetermine outer convex polygon for each block Recursively reply these steps for each blockRecursively reply these steps for each block Add to convex drawing of G all remove vertices vAdd to convex drawing of G all remove vertices v

Extendible facial cycleExtendible facial cycle Finding all facial cycle’sFinding all facial cycle’s

Convex-DrawingConvex-Drawing Algorithm convex-drawing(G, C*Algorithm convex-drawing(G, C*00))

Step 1: Step 1: V V ≥ 4, no single cycle≥ 4, no single cyclechoose v choose v C*C*00

G’ := G – vG’ := G – vdivide G’ into blocks Bdivide G’ into blocks Bii, 1 , 1 ≤ i ≤≤ i ≤ p pvv1 1 , v, vp+1p+1 outer vertices adjacent to v outer vertices adjacent to vvvii 2 2 ≤ i ≤≤ i ≤ p cut vertex of G’ s.t. v p cut vertex of G’ s.t. vi i = V(B= V(Bi-1i-1) ) V(BV(Bii))

Condition ICondition I For each inner vertex v with d(v) For each inner vertex v with d(v) ≥≥ 3, 3,

there exists three paths disjoint except there exists three paths disjoint except v, each joining v and an outer vertexv, each joining v and an outer vertex

G – V(CG – V(Coo(G)) has no connected (G)) has no connected component H such that all the outer component H such that all the outer vertices adjacent to vertices in H lie on vertices adjacent to vertices in H lie on a single path Pa single path Pii are joined by an inner are joined by an inner edgeedge

Any cycle containing no outer edge has Any cycle containing no outer edge has at least three vertices of degree at least three vertices of degree ≥ ≥ 33

Convex-DrawingConvex-Drawing Step 2:Step 2:

find a convex drawing of each block Bfind a convex drawing of each block Bii Step 2.1Step 2.1

determine outer convex polygon C*i:determine outer convex polygon C*i:locate vertices in C*locate vertices in C*0 0 – G– G00(G) in interior of (G) in interior of triangle v, vtriangle v, vii, v, vi+1 i+1 s.t. vertices adjacent to s.t. vertices adjacent to v are apicesv are apices

Step 2.2Step 2.2recursively call convex-drawing(Brecursively call convex-drawing(Bii, C*, C*ii))

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