Convex drawing chapter 5

1. Convex drawingchapter 5 Ingeborg Groeneweg

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

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

4. Facial cycle • Boundary of a face • Facial cycle of a graph G is boundary of outer face • Facial cycle of G also called C0(G) • C*0, outer convex polygon, polygonal drawing of C0

5. Extendible • C*0 is extendible if there exists a convex drawing of G with C0(G) drawn as C*0 • Let C*0 be a k-gon, k ≥ 3 • P1, P2, .. ,Pk paths in C0(G), corresponding to a side of C*0 • C*0 is extendible if and only if Condition I holds

6. 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 • G – V(Co(G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path Pi are joined by an inner edge • Any cycle containing no outer edge has at least three vertices of degree ≥ 3

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

8. Separation pair • Two subgraphs G1 = (V1, E1) , G2 = (V2,E2) of 2-connected graph G = (V,E) • (x,y)  V is separation pair if • V = V1  V2 , {x,y}= V1  V2 • E = E1  E2 ,  = E1  E2 • E1 ≥ 2 , E2 ≥ 2

9. Separation pair • example

10. Split graphs • Split graphs: obtained by adding a virtual edge (x,y) to G1 and G2 • Splitting: dividing graph into two split graphs • Split components: splitting (split) graphs until no more splits are possible

11. 3-connected components • merging split components • Triple bonds into a bond • Triangles into a ring(= a cycle) • 3-connected components are unique 1 1 1 2 3 3 2 3 2 6 7 2 3 3 2 4 5 4 5

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

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

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

15. Condition II • Let C*0 be outer strict convex polygon • 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

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

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

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

19. Extendible facial cycle • Finding all facial cycle’s

20. Convex-Drawing • Algorithm convex-drawing(G, C*0) • Step 1: V ≥ 4, no single cyclechoose v  C*0G’ := G – vdivide G’ into blocks Bi, 1 ≤ i ≤ pv1 , vp+1 outer vertices adjacent to vvi 2 ≤ i ≤ p cut vertex of G’ s.t. vi = V(Bi-1)  V(Bi)

21. 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 • G – V(Co(G)) has no connected component H such that all the outer vertices adjacent to vertices in H lie on a single path Pi are joined by an inner edge • Any cycle containing no outer edge has at least three vertices of degree ≥ 3

22. Convex-Drawing • Step 2:find a convex drawing of each block Bi • Step 2.1determine outer convex polygon C*i:locate vertices in C*0 – G0(G) in interior of triangle v, vi, vi+1 s.t. vertices adjacent to v are apices • Step 2.2recursively call convex-drawing(Bi, C*i)

23. Questions