Download
1 / 33
330 likes | 472 Views
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 ).
E N D
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 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.
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.
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.
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.
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.
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).
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.
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.
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
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.
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
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
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).
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.
Forbidden Subgraph • Theorem: The following graph is nota k-EPT graph, for any fixed k > 1.
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.
D is k-EPT k2 is not 1-EPT Graph D
[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)
Subtrees of a Tree (,,1) (i) (ii) (iii) McMorris & Scheinerman 1991 (iv) (v) Jamison & Mulder 2000
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.
Subtrees of a Tree (,,1) (i) (ii) (iii) McMorris & Scheinerman 1991 (iv) (v) Jamison & Mulder 2000
Orthodox Representations orth(,2,1) orth(3,2,1) orth(3,2,2) Theorem 6.8
The Complete Heirarchy 6.4 6.8 6.7
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) ___ ___ ___ ___ ___
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
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
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 nd + 1.
