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Twenty Years of EPT Graphs: From Haifa to Rostock

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. The Story Begins. Bell Labs in New Jersey (Spring 1981)

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Twenty Years of EPT Graphs: From Haifa to Rostock

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  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. The Story Begins 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?

  3. The Story Begins 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?

  4. The Story Begins 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? An Olive Tree Network • 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.

  5. Vertex Intersection Graphs of Paths in a Tree (VPT)Edge Intersection Graphs of Paths in a Tree (EPT) Representation P of paths in a tree T. a d b c Pc GVPT(P) GEPT(P) Each vertex v in V(GVPT) and V(GEPT) corresponds to a path Pv in T. (x,y)EVPT paths Px and Py intersect on at least one vertex in T. (x,y)EEPT  paths Px and Py intersect on at least one edge in T.

  6. Vertex and Edge Intersections of Paths For VPT-representation Pa and Pb intersect.For EPT-representation: Paand Pb do not intersect. For both VPT-representation and EPT-representation Pa and Pb intersect.

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

  8. VPT Graphs are ChordalEPTGraphs 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 C6 called a “6-pie”. 1 6 2 5 3 4 Chordless cycles have a unique EPT representation.

  10. A First Observation (Cycles) An EPT representation of C6 called a “6-pie”. 1 6 2 Theorem (Golumbic Jamison 1985): Let P be an EPT representation of G. If G contains a chordless cycle Cm (m4), then P contains an m-pie representing the cycle. 5 3 4 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 Example.The graph C6 requires degree 5. d a a b d a c b d c C6 c b A 4-pie on a,b,c,d

  14. Restricting the degree of the host tree Example.The graph C6 requires degree 5. x d a x y a b x y d a c b d c C6 c b y Now add x and y

  15. A Second Observation (Cliques) Two EPT representations of K6 called a “claw clique” or “edge clique”. All share some edge of the claw 6 1 2 5 3 4 All share a common edge

  16. A Second Observation (Cliques) Two EPT representations of K6 called a “claw clique” or “edge clique”. 6 1 2 5 Theorem (Golumbic Jamison 1985): Let P be an EPT representation of G. If G contains a clique Km (m3), then P contains either a claw or edge for it. 3 4 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. a e a e Create dummy nodes and shortenaande b b d d c c

  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. • Assume the no kissing lemma, and let U denote all paths passing through u. • Let x be a simplicial vertex of the induced subgraph GU . Thus, all paths in U share an edge with Px . • Perform the transformation: • Repeat for all nodes until max degree is 3. z z x x y y

  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 Degree 4 host trees • Theorem (2005, Golumbic, Lipshteyn, Stern): • weakly chordal  EPT  deg4 EPT

  24. Weakly Chordal Graphs Definition Weakly Chordal Graph No induced Cm for m 5, and no induced Cm 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.

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

  26. Degree 4 Trees 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. Sketch of the proof. () By the Pie Theorem, G has no induced Cm (m  5) nor C5 (=C5). By our earlier example, C6 requires degree 5. By a theorem of Golumbic and Jamison 1985, Cm (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 GU is weakly chordal, so there are two cases: • GU is a clique or GU has a two-pair.

  28. Degree 4 Trees (continued) Case 1.GU is a clique. If it were a claw clique, then u would have degree 3, and we are done. Otherwise, GU is an edge clique, and all paths in U share an edge, say (v1,u). Perform the transformation: v2 v2 v1 v1

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

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

  31. Degree 4 Trees (continued) S Subcase 2b.GS is not a clique. There exists a path Pv (vU \ S) that contains only one of the edges (v1,u),(v2,u), (v3,u),(v4,u), say (v1,u). Let  be the non-empty collection of such paths, which thus form an edge clique containing (v1,u). Color the star edges as follows:

  32. Subcase 2b, continued. 1. (v1,u) is not colored. 2. (vi,u), i  5 is colored pink if it is contained in a path in . 3. (vi,u) is colored if is contained in a path that already has a pink edge. Example. [v1,u,v5]  P1 [v1,u,v7]  P2 [v5,u,v6]  P3  = {P1,P2} Lemma: The edges of Py are not colored. Q.E.D.

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

  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 deg3EPTordeg4EPTgraphs, 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 Tx and Ty share t vertices

  36. [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 Tx and Ty share t vertices Interval graphs  [2,2,1] EPT  [, 2, 2] Chordal graphs  [,,1]  [3,3,1]  [3,3,2] (MS,JM)

  37. [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 Tx and Ty share t vertices 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)

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

  39. A New Characterization Theorem Golumbic, Lipshteyn, Stern [WG2006]: The class [4,4,2]-graphs is equivalent to weakly chordal (K2,3, P6, 4P2, P2  P4 , H1, H2 , H3)-free. K1and K2 are cliques of size at most 2.

  40. Forbidden Subgraphs

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

  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. • 1-EPT k-EPT, for any fixed k > 1. • When restricted to degree 3 trees, the containment is also strict.

  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-EPTImportant 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-claw clique k-edge clique

  46. Recognition of k-EPTBranch Graphs Definition: Let C be a subset of vertices of G. The branch graph B(G/C): B(G/C) G 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-EPTNP-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. Pijpath in T (i,j) E(H) Qiedge in T i V(H) T H

  48. T Pijpath in T (i,j) E(H) H Qiedge in T i V(H) {Pij} 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

  49. Recognition of k-EPTCorollaries • 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.

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