1 / 33

The k -Edge Intersection Graphs of Paths in a Tree

The k -Edge Intersection Graphs of Paths in a Tree. Martin Charles Golumbic Marina Lipshteyn Michal Stern Caesarea Rothschild Institute, University of Haifa, Israel. Definition of V ertex Intersection Graphs of P aths in a T ree ( VPT ).

trung
Download Presentation

The k -Edge Intersection Graphs of Paths in a Tree

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The k-Edge Intersection Graphs of Paths in a Tree Martin Charles Golumbic Marina Lipshteyn Michal Stern Caesarea Rothschild Institute, University of Haifa, Israel

  2. Definition of Vertex Intersection Graphs of Paths in a Tree (VPT) Gavril (1978): VPT graphs (i.e., path graphs) are chordal (no induced cycle Ck.) VPT representation tree T of G G = (V,E) Pc Each vertex v in V(G) corresponds to a path Pv in T. (x,y) E paths Px and Py intersect on at least one vertex in T. Def. G is a VPT graph if G has a VPT representation.

  3. VPT Graphs are Chordal Gavril (1978): Chordal graphs are equivalent to the vertex intersection graphs of subtrees of a tree. Def. A graph G is a Chordal graph if G has no chordless cycle Ck for k  4. Therefore: VPT graphs (i.e., path graphs) are chordal.

  4. Definition of k-Edge Intersection Graphs of Paths in a Tree (k-EPT) k-EPT representation tree T of G G = (V,E) Remark. k-EPT are also known as tolerance intersection graphs of paths in a tree with constant tolerance k+1. Each vertex v in V(G) corresponds to a path Pv in T. (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.

  5. 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.

  6. Applications • Scheduling undirected calls in a tree network. Conflicting paths require distinct time slots (color the graph) • Assigning wavelengths to connections in an optical network.

  7. Known Results (for k = 1) • The recognition and coloring problems of a 1-EPT graph are NP-complete(Golumbic and Jamison, 1985). • Tarjan (1985) gave a 3/2-approximation algorithm for coloring 1-EPT graphs. • VPT graphs are incomparable with 1-EPT graphs, but • On degree 3 trees VPT graphs equal 1-EPT graphs (Golumbic and Jamison, 1985).

  8. 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.

  9. 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 > 1.

  10. 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

  11. 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.

  12. 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

  13. 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

  14. 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).

  15. 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.

  16. Forbidden Subgraph • Theorem: The following graph is nota k-EPT graph, for any fixed k > 1.

  17. 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 k-EPT but are a (k+1)-EPT graph, for any fixed k.

  18. C4 is not VPT but is k-EPT k1

  19. D is k-EPT k2 is not 1-EPT Graph D

  20. [h,s,p] 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 p: An edge (i,j) in G if Ti and Tj share p vertices [, 2, k+1] is k-EPT [3, 2, 2] is deg 3, 1-EPT [, 2, 1] is VPT [3,3,3] in Jamison-Mulder 2000 [, , 1] is chordal graphs[, , 2] are all graphs (i.e., intersection graphs of subtrees of a tree)

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

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

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

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

  25. The Complete Heirarchy

  26. The Complete Heirarchy 6.4 6.8 6.7

  27. Chordal and Weakly Chordal Graphs • Chordal : No induced Cn for n 4. • Weakly Chordal : No induced Cn for n 5, and no induced Cn for n 5. C4 , C5 , C6, C7 , … are k-EPT (k 1) C5 , C6 are k-EPT (k 1) C7 , C8 , … are not k-EPT (k 1) ___ ___ ___ ___ ___

  28. Chordal 1-EPT Graphs Golumbic and Jamison 1985 The following are equivalent: - G is chordal and 1-EPT - G has a degree 3, 1-EPT representation

  29. Weakly Chordal 1-EPT Graphs Golumbic, Lipshteyn and Stern 2005 The following are equivalent: - G is weakly chordal and 1-EPT - G has a degree 4, 1-EPT representation

  30. The Complete Heirarchy

  31. Open Question For which values of d does the following hold? A 1-EPT graph G has a degree d representation if and only if G has no induced Cn for nd + 1.

  32. More on Algorithmic Graph Theory

  33. The Endof the Talk.Thank you!

More Related