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Convex drawing chapter 5 Ingeborg Groeneweg

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Summery What is convex drawing What is convex drawing Some definitions Some definitions Testing convexity Testing convexity Drawing a convex graph Drawing a convex graph

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Convex drawing Drawing is called convex: Drawing is called convex: Each egde straight line Each egde straight line Each face convex polygon Each face convex polygon Not every planar graph is convex Not every planar graph is convex Every 3-connected planar graph has a convex drawing Every 3-connected planar graph has a convex drawing

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Facial cycle Boundary of a face Boundary of a face Facial cycle of a graph G is boundary of outer face Facial cycle of a graph G is boundary of outer face Facial cycle of G also called C 0 (G) Facial cycle of G also called C 0 (G) C* 0, outer convex polygon, polygonal drawing of C 0 C* 0, outer convex polygon, polygonal drawing of C 0

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Extendible C* 0 is extendible if there exists a convex drawing of G with C 0 (G) drawn as C* 0 C* 0 is extendible if there exists a convex drawing of G with C 0 (G) drawn as C* 0 Let C* 0 be a k-gon, k ≥ 3 Let C* 0 be a k-gon, k ≥ 3 P 1, P 2,..,P k paths in C 0 (G), corresponding to a side of C* 0 P 1, P 2,..,P k paths in C 0 (G), corresponding to a side of C* 0 C* 0 is extendible if and only if Condition I holds C* 0 is extendible if and only if Condition I holds

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Condition I For each inner vertex v with d(v) ≥ 3, there exists three paths disjoint except v, each joining v and an outer vertex For each inner vertex v with d(v) ≥ 3, there exists three paths disjoint except v, each joining v and an outer vertex G – V(C o (G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path P i are joined by an inner edge G – V(C o (G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path P i are joined by an inner edge Any cycle containing no outer edge has at least three vertices of degree ≥ 3 Any cycle containing no outer edge has at least three vertices of degree ≥ 3

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Definitions Separation pair Separation pair Split component Split component 3-connected component 3-connected component Prime separation pair Prime separation pair Forbidden separation pair Forbidden separation pair Critical separation pair Critical separation pair

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Separation pair Two subgraphs G 1 = (V 1, E 1 ), G 2 = (V 2,E 2 ) of 2-connected graph G = (V,E) Two subgraphs G 1 = (V 1, E 1 ), G 2 = (V 2,E 2 ) of 2-connected graph G = (V,E) (x,y) V is separation pair if (x,y) V is separation pair if V = V1 V2, {x,y}= V1 V2 V = V1 V2, {x,y}= V1 V2 E = E1 E2, = E1 E2 E = E1 E2, = E1 E2 E1 ≥ 2, E2 ≥ 2 E1 ≥ 2, E2 ≥ 2

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Separation pair example example

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Split graphs Split graphs: obtained by adding a virtual edge (x,y) to G 1 and G 2 Split graphs: obtained by adding a virtual edge (x,y) to G 1 and G 2 Splitting: dividing graph into two split graphs Splitting: dividing graph into two split graphs Split components: splitting (split) graphs until no more splits are possible Split components: splitting (split) graphs until no more splits are possible

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3-connected components merging split components merging split components Triple bonds into a bond Triple bonds into a bond Triangles into a ring(= a cycle) Triangles into a ring(= a cycle) 3-connected components are unique 3-connected components are unique 1 2 3 4 5 6 7 2 1 1 3 2 2 3 3 4 5 2 3

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Prime separation pair Prime separation pair {x,y}: Prime separation pair {x,y}: x and y end vertices of virtual edge contained in 3- connected component x and y end vertices of virtual edge contained in 3- connected component 1 2 3 4 5 6 7 2 3 2 2 3 3 4 5 2 3

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Forbidden separation pair Prime separation pair is forbidden if: Prime separation pair is forbidden if: At least four {x,y}-split components, or At least four {x,y}-split components, or Exactly three {x,y}-split components: no ring, no bond Exactly three {x,y}-split components: no ring, no bond

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Critical separation pair Prime separation pair is critical if: Prime separation pair is critical if: Exactly three {x,y}-split components including a ring or a bond, or Exactly three {x,y}-split components including a ring or a bond, or Exactly two {x,y}-split components: no ring, no bond Exactly two {x,y}-split components: no ring, no bond = =

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Condition II Let C* 0 be outer strict convex polygon Let C* 0 be outer strict convex polygon G has no forbidden separation pair G has no forbidden separation pair For each critical separation pair (x,y) of G, there is at most one (x,y)-split component having no edge of F, and if any, it is either a bond if (x,y) E or a ring otherwise For each critical separation pair (x,y) of G, there is at most one (x,y)-split component having no edge of F, and if any, it is either a bond if (x,y) E or a ring otherwise

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Testing convexity Forbidden separtion pair no convex drawing Forbidden separtion pair no convex drawing No forbidden, one critical convex drawing some outer facial cycle No forbidden, one critical convex drawing some outer facial cycle No forbidden, two or more critical further specification No forbidden, two or more critical further specification No forbidden, No critical convex drawing for any facial cycle, subdivision of 3-connected graph No forbidden, No critical convex drawing for any facial cycle, subdivision of 3-connected graph

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Testing convexity For every critical separation pair (x,y) For every critical separation pair (x,y) (x,y) E -> delete (x,y) (x,y) E -> delete (x,y) (x,y) E and one (x,y)-split component is ring -> delete x-y path (x,y) E and one (x,y)-split component is ring -> delete x-y path Resulting graph G’ Resulting graph G’ Add to G’ new vertex v Add to G’ new vertex v Join v to all critical separation vertices Join v to all critical separation vertices If new Graph G’’ is planar G has convex drawing If new Graph G’’ is planar G has convex drawing

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

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Extendible facial cycle Finding all facial cycle’s Finding all facial cycle’s

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Convex-Drawing Algorithm convex-drawing(G, C* 0 ) Algorithm convex-drawing(G, C* 0 ) Step 1: V ≥ 4, no single cycle choose v C* 0 G’ := G – v divide G’ into blocks B i, 1 ≤ i ≤ p v 1, v p+1 outer vertices adjacent to v v i 2 ≤ i ≤ p cut vertex of G’ s.t. v i = V(B i-1 ) V(B i ) Step 1: V ≥ 4, no single cycle choose v C* 0 G’ := G – v divide G’ into blocks B i, 1 ≤ i ≤ p v 1, v p+1 outer vertices adjacent to v v i 2 ≤ i ≤ p cut vertex of G’ s.t. v i = V(B i-1 ) V(B i )

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Condition I For each inner vertex v with d(v) ≥ 3, there exists three paths disjoint except v, each joining v and an outer vertex For each inner vertex v with d(v) ≥ 3, there exists three paths disjoint except v, each joining v and an outer vertex G – V(C o (G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path P i are joined by an inner edge G – V(C o (G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path P i are joined by an inner edge Any cycle containing no outer edge has at least three vertices of degree ≥ 3 Any cycle containing no outer edge has at least three vertices of degree ≥ 3

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Convex-Drawing Step 2: find a convex drawing of each block B i Step 2: find a convex drawing of each block B i Step 2.1 determine outer convex polygon C*i: locate vertices in C* 0 – G 0 (G) in interior of triangle v, v i, v i+1 s.t. vertices adjacent to v are apices Step 2.1 determine outer convex polygon C*i: locate vertices in C* 0 – G 0 (G) in interior of triangle v, v i, v i+1 s.t. vertices adjacent to v are apices Step 2.2 recursively call convex-drawing(B i, C* i ) Step 2.2 recursively call convex-drawing(B i, C* i )

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