1. Circle Graph and Circular Arc Graph Recognition

2. 2/41 Outlines Circle Graph Recognition Circular-Arc Graph Recognition

3. 3/41 Chords of a circle A chord v is associated two endpoints {x,y} a e f d b c x y

4. 4/41 Circle Graph A circle graph is a graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords in the circle intersect. e a d b f c a e f d b c

5. 5/41 Unique Chord Models Consider the two equivalent Chord Models of the following circle graph. A circle graph is uniquely represented if all of its Chord Models are equivalent. Shift and Reverse

6. 6/41 Circle Graph Recognition

7. 7/41 Overview Let G=(V,E) be the input circle graph. Decompose a given graph G into the Prime Subgraphs through join decomposition –Each prime component has a unique chord model Create a a unique chord model for a Prime Subgraph –Find a member W of a family F whose members have unique chord models first. –Determine the chords for the vertices in V-W.

8. 8/41 Join Decomposition V1V1 V3V3 V2V2 V0V0 All Edges m2m2 m1m1 V3V3 V2V2 m2m2 V0V0 V1V1 m1m1 G1G1 G2G2

9. 9/41 Prime Subgraphs A subgraph G’=(V’, E’) is a prime subgraph iff it is join-inseperatable. A prime subgraph G’, |V’|≥ 5, contains a subgraph (a member of the set F) which has a simple and unique chord presentation. The family of the subgraph which has a simple and unique chord presentation is named as the set F.

10. 10/41 Members of the Set F (a) a house (b) A tepee (c) A figure-8 (d) Primitive cycles of length k ≧ 5

11. 11/41 The Members of the Set F 2 345 1 (b) A tepee 1 2 3 4 5 Round tent

12. 12/41 The unique placement of chords Assume we have found a W which is a member of f. We construct the unique chord model for W first. Then, if we arrange the remaining vertices carefully, there is a unique placement for each of them iteratively. ( C. Gabor, W. L. Hsu and K. Supowit, Recognizing circle graphs in polynomial time," J. Assoc. Comput. Machin., 435-473, (1989) )

13. Circular-Arc Graph Recognition Check the flow chart in the handout

14. 14/41 Circular-Arc Graph A circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect. 3 2 8 7 6 4 5 1

15. 15/41 An Arc Model Denote arc i by head(i), tail(i) scanned in the clockwise order.

16. 16/41 Two Different Arc Models unnecessary variations 1 2 3 4 5 4 2 3 1 5 4 2 3 1 5

17. 17/41 Vertex Adjacency in circular arc graphs Two vertices v 1 and v 2 in G are said to be –Independent if v 1 is not adjacent to v 2. –strictly adjacent if v 1 is adjacent to v 2 but neither N(v 1 ) nor N(v 2 ) is contained in the other. –strongly adjacent if v 1 and v 2 are strictly adjacent and every w in V(G)\N(v 1 ) satisfies that N(w) ⊆ N(v 2 ) and every w’ in V(G)\N(v 2 ) satisfies that N(w’) ⊆ N(v 1 ). –Similar if N(v 1 )\{v 1 }=N(v 2 )\{v 2 } To define normalized models, we need to assume there are no similar vertices

18. 18/41 Overlapping relationships of a pair of arcs

19. 19/41 A normalized Arc Model An arc Model R for a Circular-Arc Graph G is normalized if every pair of arcs u 1 and u 2 and the corresponding pair of nodes v 1 and v 2 satisfy –(1) u 1 is independent of u 2 v 1 is not adjacent to v 2 –(2) u 1 is contained in u 2 N(v 1 ) ⊆ N(v 2 ) –(3) u 1 strictly overlaps u 2 v 1 is strictly but not strongly adjacent to v 2 –(4) u 1 and u 2 cover the circle v 1 is strictly but not strongly adjacent to v 2

20. 20/41 A Unique Normalized Model A first step towards unique arc model. –To avoid unnecessary model variation, define a Normalized Model for each circular-arc graph An arc model C is equivalent to C’ if C’ can be obtained from C by rotation (shift) and reflection (clockwise → counter-clockwise). A circular-arc graph is said to have a unique normalized model if all normalized models are equivalent.

21. 21/41 How to normalize an existing arc model? Possible Violations Type I u 1 strictly overlaps u 2 but v 1 and v 2 are strongly adjacent. (Algorithm I) Type II u 1 strictly overlaps u 2 but v 1 is contained in v 2. (Algorithm II)

22. 22/41 Two types of Violations If R is not a normalized model for G, there are two types of violations. Type I u 1 strictly overlaps u 2 but v 1 and v 2 are strongly adjacent. (eliminated by Algorithm I) Type II u 1 strictly overlaps u 2 but v 1 is contained in v 2. (eliminated by Algorithm II)

23. 23/41 Endpoint Blocks of an Arc Model A head (or tail) block is a set of maximal contiguous subsequence of heads (tails) 12341234 h(4) h(1) h(3) h(2) t(4) t(1) t(3) t(2) head block H tail block T

24. 24/41 Algorithm I (eliminating Type I violation) 1 For each head h(i), find the first tail block T(h(i)) by a counterclockwise traversal from h(i) h(i) t(i) T(h(i))

25. 25/41 Algorithm I (eliminating Type I violation) 2 T(h(i)) = T 1 ∪ T 2. T 1 is the tails of those corresponding arcs not overlapping arc i. T 2 = T(h(i))\T 1 h(i) t(i) T(h(i)) T2 T1

26. 26/41 Algorithm I (eliminating Type I violation) 3 If T 2 is not empty, insert h(i) between T 2 and T 1. Repeat the procedure for t(i). h(i) t(i) T2T2 T1T1

27. 27/41 Algorithm II (eliminating Type II violation) For each head block, sort the order of the heads by its corresponding reverse order of tails. 12341234 h(4) h(1) > h(3) t(4) > t(3) > t(2) > t(1)

28. 28/41 The Associated Circle Graph G c Associate with each graph G the graph G c that has the same vertex set as G such that two vertices in G c are adjacent iff they are strictly but not strongly adjacent in G. G GcGc

29. 29/41 Utilizing the Associated Circle Graph G c G is a circular-arc graph iff all induced subgraphs of G are circular arc graph. Use the Associated Circle Graph G c to construct a normalized arc model of G. However, the same arc model may still associate with multiple chords model. –How to deal with such a phenomenon? Graph decomposition

30. 30/41 The non-Unique Chord Representation (Type I) Consider the connected components of G c

31. 31/41 Consider the “module-free” subgraph of G c The non-Unique Chord Representation (Type II)

32. 32/41 Modular Decomposition To obtain components which do not contain type I or type II structures, can use Modular Decomposition to decompose G c

33. 33/41 Conformal Model Does the chord model necessarily represent a N-model? The chord model represents a N-model if it is a Conformal Model. Three chords d 1, d 2 and d 3 (d 1 | d 2 | d 3 ) are said to be parallel if d1d1 d2d2 d3d3 d1d1 d2d2 d3d3 parallel series

34. 34/41 Conformal Model Three nodes v i,v j and v k of G are parallel (v i |v j |v k ) if one of the following 8 conditions is satisfied –(1) v k contains v j v j contains v i. –(2) v k contains v j ; v j and v i cover the circle. –(3) v k is contained in v j ; v i contains v j. –(4) v k is contained in v j ; v i is independent of v j. –(5) v k is independent of v j ; v j contains v i. –(6) v k is independent of v j ; v j and v i cover the circle. –(7) v k and v j cover the circle; v j is independent of v i –(8) v k and v j cover the circle; v i contains v j.

35. 35/41 Conformal Model A chord model is a Conformal Model iff v 1 |v 2 |v 3 when d 1 |d 2 |d 3. Let G be a circular-arc graph with a connected G c ’ (the complement of G c ). Then a model for G c is conformal iff it is a chord model associated with an N-model for G.

36. 36/41 The Transformation from chords back to Arcs (easy part) Let G be a circular-arc graph with a connected G c ’ (the complement of G c ). Let D be an associated chord model of G c. Then D determines a unique N-model R of G. For a fixed arc i, a new arc j –Is on the opposite side of i if i is independent of j –or cover the circle with j. A new arc j on the same side of i if i contains j or is contained in j.

37. 37/41 The Transformation from chords back to Arcs 2 1 3 4 5 6 7 2 1 3 4 5 6 7 1 5 2 3 4 6 7 Gc The complement of Gc

38. 38/41 Summary Recognition Algorithm –Take a proposed circular-arc graph G and transform it into a circle graph G c –Decompose G c into s-inseparable components. –If G is indeed a circular-arc graph, we can find a conformal model of G c and transform it back to an N-model of G. –Otherwise, there is a contradiction, and G is not a circular arc graph.