1. Twenty Years of EPT Graphs: From Haifa to Rostock Martin Charles Golumbic Caesarea Rothschild Institute University of Haifa With thanks to my research collaborators: Robert Jamison, Marina Lipshteyn, Michal Stern

2. Bell Labs in New Jersey (Spring 1981) John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls? The Story Begins

3. Bell Labs in New Jersey (Spring 1981) John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls? The Story Begins

4. Bell Labs in New Jersey (Spring 1981) John Klincewicz: Suppose you are routing phone calls in a tree network. Two calls interfere if they share an edge of the tree. How can you optimally schedule the calls? The Story Begins A call is a path between a pair of nodes. A typical example of a type of intersection graph. Intersection here means “share an edge”. Coloring this intersection graph is scheduling the calls. An Olive Tree Network

5. Vertex Intersection Graphs of Paths in a Tree (VPT) Edge Intersection Graphs of Paths in a Tree (EPT) Each vertex v in V(G VPT ) and V(G EPT ) corresponds to a path P v in T. (x,y)  E VPT  paths P x and P y intersect on at least one vertex in T. (x,y)  E EPT  paths P x and P y intersect on at least one edge in T. Representation P of paths in a tree T. G VPT ( P ) PcPc G EPT ( P ) a d b c

6. For VPT-representation P a and P b intersect. For EPT-representation: P a and P b do not intersect. For both VPT-representation and EPT-representation P a and P b intersect. Vertex and Edge Intersections of Paths

7. Theorem. Chordal graphs  vertex intersection graphs of subtrees of a tree. [Buneman], [Gavril], [Walter] A graph G is chordal if every cycle of size  4 has a chord, i.e., G has no induced chordless cycles C m for m  4. Chordal Graphs

8. VPT Graphs are Chordal EPT Graphs are Not Chordal A path is a subtree, therefore VPT graphs (i.e., path graphs) are chordal. However, EPT graphs may have chordless cycles of any size.

9. A First Observation (Cycles) An EPT representation of C 6 called a “6-pie”. 6 3 2 1 4 5 Chordless cycles have a unique EPT representation.

10. A First Observation (Cycles) An EPT representation of C 6 called a “6-pie”. 6 3 2 1 4 5 Theorem (Golumbic Jamison 1985): Let P be an EPT representation of G. If G contains a chordless cycle C m (m  4), then P contains an m-pie representing the cycle. Chordless cycles have a unique EPT representation.

11. Restricting the degree of the host tree Remark. If m is the maximum degree in T, then the EPT graph has no chordless (m+1) cycles (or larger).

12. Restricting the degree of the host tree Remark. If m is the maximum degree in T, then the EPT graph has no chordless (m+1) cycles (or larger). Corollary: If P is an EPT representation of G on a degree 3 tree T. Then G is chordal graph.

13. Restricting the degree of the host tree C6C6 a a a b b b c c c d d d Example. The graph C 6 requires degree 5. A 4-pie on a,b,c,d

14. Restricting the degree of the host tree C6C6 a a a b b b c c c d d d x x x y y y Example. The graph C 6 requires degree 5. Now add x and y

15. A Second Observation (Cliques) Two EPT representations of K 6 called a “claw clique” or “edge clique”. 6 3 2 1 4 5 edge All share a common edge claw All share some edge of the claw

16. A Second Observation (Cliques) Two EPT representations of K 6 called a “claw clique” or “edge clique”. 6 3 2 1 4 5 Theorem (Golumbic Jamison 1985): Let P be an EPT representation of G. If G contains a clique K m (m  3), then P contains either a claw or edge for it. Cliques have exactly two possible EPT representations.

17. No Kissing The No Kissing Lemma: If P is an EPT representation of G on a tree T, and u is any node of T. We may assume without loss of generality, and without increasing the degree of the tree, that all paths touching u continue through u. No stopping. No kissing u.

18. No Kissing The No Kissing Lemma: If P is an EPT representation of G on a tree T, and u is any node of T. We may assume without loss of generality, and without increasing the degree of the tree, that all paths touching u continue through u. No stopping. No kissing u. e c b a d e c b a d Create dummy nodes and shorten a and e

19. Degree 3 host trees If P is a deg3 EPT representation of G on a tree T, then applying the No Kissing Lemma construction to all nodes of degree 3 yields a deg3 VPT representation of G.

20. Degree 3 host trees If P is a deg3 EPT representation of G on a tree T, then applying the No Kissing Lemma construction to all nodes of degree 3 yields a deg3 VPT representation of G. i.e., deg3 EPT  deg3 VPT  deg3 EPT  chordal  EPT Now let’s prove: chordal  EPT  deg3 EPT

21. Degree 3 host trees (continued) Let P be any EPT representation of G on a tree T, and u any node of T of maximum degree d > 3. 1.Assume the no kissing lemma, and let U denote all paths passing through u. 2.Let x be a simplicial vertex of the induced subgraph G U. Thus, all paths in U share an edge with P x. 3.Perform the transformation: 4.Repeat for all nodes until max degree is 3. x y z x y z

22. Degree 3 host trees (continued) Theorem (1985): All four classes are equivalent: chordal  EPT  deg3 EPT  VPT  EPT  deg3 VPT What about degree 4?

23. Degree 3 host trees (continued) Theorem (1985): All four classes are equivalent: chordal  EPT  deg3 EPT  VPT  EPT  deg3 VPT Theorem (2005, Golumbic, Lipshteyn, Stern): weakly chordal  EPT  deg4 EPT Degree 4 host trees

24. Definition Weakly Chordal Graph No induced C m for m  5, and no induced C m for m  5. Theorem [Hayward, Hoàng, Maffray 1989] G is weakly chordal if and only if every induced subgraph of G is either a clique or has a two-pair. Weakly Chordal Graphs

25. A two-pair is a pair of vertices, such that every chordless path between them has length two edges. Remark. If {x,y} is a two-pair, then the common neighborhood of x and y is an (x,y)-minimal separator. {x,y} is a two-pair{x,y} is not a two-pair

26. Theorem [GLS 2005] A graph G has an EPT representation on a degree 4 tree if and only if G is a weakly chordal EPT graph. Degree 4 Trees Sketch of the proof. (  ) By the Pie Theorem, G has no induced C m (m  5) nor C 5 (=C 5 ). By our earlier example, C 6 requires degree 5. By a theorem of Golumbic and Jamison 1985, C m (m  7) is not an EPT graph.

27. Degree 4 Trees (continued) (  ) Let P be an EPT representation of G on tree T with maximal degree d > 4, and let u be a node of degree d. We transform P into an EPT representation P´ on T´ with fewer vertices of degree d. The full proof follows by induction. Assume the no kissing lemma at u, and let U denote all paths passing through u. The induced subgraph G U is weakly chordal, so there are two cases: G U is a clique or G U has a two-pair.

28. Degree 4 Trees (continued) Case 1. G U is a clique. If it were a claw clique, then u would have degree 3, and we are done. Otherwise, G U is an edge clique, and all paths in U share an edge, say (v 1,u). Perform the transformation: v1v1 v1v1 v2v2 v2v2

29. Degree 4 Trees (continued) Case 2. G U has a two pair {x,y}. The common neighborhood S of {x,y} is a minimal separator and splits G U into (at least) 2 connected components: G X containing x and G Y containing y. The star edges centered at u, are now painted. The two contained in P x are red; those in P y blue. Propagate the coloring to other star edges via the paths P z (z  U \ S): if P z has one red edge, then paint its other star edge red; if P z has one blue edge, then paint its other star edge blue. No star edges gets two colors!

30. Degree 4 Trees (continued) Subcase 2a. G S is a clique. Perform the transformation: S

31. Degree 4 Trees (continued) Subcase 2b. G S is not a clique. There exists a path P v (v  U \ S) that contains only one of the edges (v 1,u),(v 2,u), (v 3,u),(v 4,u), say (v 1,u). Let  be the non- empty collection of such paths, which thus form an edge clique containing (v 1,u). Color the star edges as follows: S

32. Subcase 2b, continued. 1. (v 1,u) is not colored. 2. (v i,u), i  5 is colored pink if it is contained in a path in . 3. (v i,u) is colored if is contained in a path that already has a pink edge. Lemma: The edges of P y are not colored. Example. [v 1,u,v 5 ]  P 1 [v 1,u,v 7 ]  P 2 [v 5,u,v 6 ]  P 3  = {P 1,P 2 } Q.E.D.

33. From Haifa to Rostock (Spring 1985?) The Story Continues

34. Algorithmic Aspects of EPT Graphs The recognition and coloring problems of an EPT graph are NP-complete (Golumbic and Jamison, 1985). There is a 3/2-approximation algorithm for coloring EPT graphs. (Tarjan, 1985) On deg3EPT or deg4EPT graphs, coloring is polynomial since, respectively, they are chordal (GJ 1985) or weakly chordal (GLS 2005). Max-clique and Max-stable set are polynomial (GJ 1985).

35. [h,s,t] Graphs and Representations A collection of subtrees of a tree T satisfying: h:T has maximum vertex degree h s:Each subtree has maximum vertex degree s t:An edge (x,y) in G if T x and T y share t vertices

36. Interval graphs  [2,2,1] EPT  [ , 2, 2] Chordal graphs  [ , ,1]  [3,3,1]  [3,3,2] (MS,JM) [h,s,t] Graphs and Representations A collection of subtrees of a tree T satisfying: h:T has maximum vertex degree h s:Each subtree has maximum vertex degree s t:An edge (x,y) in G if T x and T y share t vertices

37. Interval graphs  [2,2,1] EPT  [ , 2, 2] Chordal graphs  [ , ,1]  [3,3,1]  [3,3,2] (MS,JM) Any graph  [ , ,2] [ ,2,2]  chordal  [3,2,2]  [3,2,1] (GJ) [ ,2,2]  weakly chordal  [4,2,2] (GLS) [h,s,t] Graphs and Representations A collection of subtrees of a tree T satisfying: h:T has maximum vertex degree h s:Each subtree has maximum vertex degree s t:An edge (x,y) in G if T x and T y share t vertices

38. Two Directions to Generalize EPT 1.Keep t = 2 (edge intersection) Increase s (generalizing to subtrees) but Bound max degree h. 2.Increase t (constant tolerence) Keep s = 2 (paths)

39. A New Characterization Theorem Golumbic, Lipshteyn, Stern [WG2006]: The class [4,4,2]-graphs is equivalent to weakly chordal  (K 2,3, P 6, 4P 2, P 2  P 4, H 1, H 2, H 3 )-free. K 1 and K 2 are cliques of size at most 2.

40. Forbidden Subgraphs

41. Definition of k-EPT Graphs The k-Edge Intersection Graphs of Paths in a Tree (x,y)  E  paths P x and P y intersect on at least k edges in T. Def. G is a k-EPT graph if G has a k-EPT representation. k-EPT representation tree T of G G = (V,E) for k = 4 edges k-EPT  [ , 2, k+1] (i.e., share k+1 vertices)

42. Examples of Intersections For VPT representation: paths a and b intersect. For k-EPT representation, k>0: paths a and b do not intersect. For VPT and 1-EPT representation: paths a and b intersect. For k-EPT representation, k>1: paths a and b do not intersect. For VPT and k-EPT representation, k 4: paths a and b intersect. For k-EPT representation, k>4: paths a and b do not intersect.

43. Properties of k-EPT 1-EPT k-EPT, for any fixed k > 1. - Divide each edge into k edges, by adding k-1 dummy vertices. When restricted to degree 3 trees, the containment is also strict. 1-EPT k-EPT, for any fixed k > 1.

44. New Properties of k-EPT VPT graphs are incomparable with k-EPT graphs, for any fixed k  1. When restricted to degree 3 trees, VPT  k-EPT, for any fixed k  2. Chordless cycles are degree 3 k-EPT for k  2.

45. Recognition of k-EPT Important Properties Any maximal clique of a k-EPT graph is either a k-edge clique or a k-claw clique. A k-EPT graph G has at most maximal cliques. k-edge clique k-claw clique

46. Recognition of k-EPT Branch Graphs Definition: Let C be a subset of vertices of G. The branch graph B(G/C): G B(G/C) Theorem: Let C be a maximal clique of a k-EPT graph G. Then the branch graph B(G/C) can be 3-colored.

47. Recognition of k-EPT NP-Completeness Theorem: It is an NP-complete problem to decide whether a VPT graph is a k-EPT graph. Proof: An arbitrary undirected graph H is 3-colorable iff a certain graph G=(V,E) is a k-EPT graph. H T P ij path in T (i,j) E(H) Q i edge in T i V(H)

48. {P ij } corresponds to a maximal clique C of the VPT graph G. B(G/C) is isomorphic to H. If G is VPT and k-EPT  H is 3-colorable If H is 3-colorable and G is VPT  G is VPT and 1-EPT  G is k-EPT. Therefore, G is VPT and k-EPT  G is VPT and H is 3-colorable T HP ij path in T (i,j) E(H) Q i edge in T i V(H)

49. Recognition of k-EPT Corollaries Corollary: Recognizing whether an arbitrary graph is a k-EPT graph is an NP-complete problem. Corollary: Let G be a VPT graph. Then G is a 1-EPT graph iff G is a k-EPT graph, (hence: iff G is chordal).

50. Coloring of k-EPT Theorem: The problem of finding a minimum coloring of a k-EPT graph is NP-complete. Same proof as Golumbic & Jamison (1985) for the case k = 1.

51. Forbidden Subgraph Theorem: The following graph is not a k-EPT graph, for any fixed k > 1.

52. Open Problem The relationships between k-EPT graphs and (k+1)-EPT graphs, for any fixed k. Is k-EPT  (k+1)-EPT? We have graphs that are not 1-EPT but are a (k+1)-EPT graph, for any fixed k.

53. C 4 is not VPT but is k-EPT k  1

54. D is k-EPT k  2 is not 1-EPT Graph D

55. Orthodox Representations P A representation for G is orthodox if For each path, its endpoints are leaves (leaf generated), and Two paths P i, P j share a leaf if and only if vertices i and j are adjacent in G.

56. Subtrees of a Tree (i)  (ii)  (iii) McMorris & Scheinerman 1991  (iv)  (v) Jamison & Mulder 2000 ( , ,1 )

57. Orthodox Representations orth( ,2,1)  orth(3,2,1)  orth(3,2,2) Theorem 6.8

58. The Complete Heirarchy

59. 6.4 6.7 6.8

60. The Complete Heirarchy

62. More on Algorithmic Graph Theory

63. Further Research Characterize families of [h,s,t] graphs for various values of h, s and t. Find intersection models as (h,s,t)-representations for known families of graphs. weakly chordal  [?, ?, ?]

64. Thank You!