Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of nodes, each two of which are joined by a path of G that avoids the neighborhood of the third node [LB62] Graphs without asteroidal triple are termed AT free graphs. This class includes, in particular, the classes of interval, permutation, bounded tolerance and co comparability graphs (see [Gol80]) It is well known that AT free graphs have a dominating pair [COS97] that can be founded in linear time. We thus have the following. O(1) for the family of AT free graphs of diameter D. Moreover, the scheme is polynomialtime constructible, and the distance decoder has O(1) time complexity. To ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, Harcourt Brace Jovanovich, Academic Press edition, 1980.

....some intersection graph families, like interval graphs, paths graphs (intersection of paths in a tree) circle graphs (intersection of chords in a circle) permutation graphs, circular arc graphs, graphs with bounded interval number . see Golumbic for de nitions of these families [Gol80] but also graphs with bounded arboricity [KNR88] The arboricity of a graph G is de ned to be k = max H jE(H)j jV (H)j 1 where H range over all possible induced subgraphs of G containing at least two nodes. Arboricity is related to degeneracy of graphs. A graph is d degenerated ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, Harcourt Brace Jovanovich, Academic Press edition, 1980.

....intersect. Interval graphs have a number of applications, for example in genetics, archeology and developmental psychology (see e.g. Rob76] Their geometric structure makes it easy to solve various optimization problems, among them nding the maximum independent set or a clique cover (see e.g. Gol80] In this paper we study how to determine, given an interval graph, an unknown independent set X chosen by an adversary. We refer to X as a hidden independent set. Note that X need not be maximal. We determine X by playing an interactive game against an adversary using queries of the following ....

....I i I j = Then either f i s j or f j s i , and, depending on this, we can orient the edge in G as i j or j i. Thus, G has a natural orientation of the edges, and this orientation is well known to be acyclic and transitive. For this and other results about interval graphs, see e.g. Gol80] Formally, an independent set of vertices of a graph G = V; E) is a subset V of V such that no two vertices in V are adjacent. We will deal with discovering an initially unknown independent set chosen by an adversary, and will refer to this set as the hidden independent set. If V is ....

Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Academic Press, New York, 1980.

.... 2 2n 3m vertices and, by Proposition 3.17, tk k(k 1) 2 edges. Therefore T can be constructed in polynomial time in the parameters n and m of any instance of the problem 3 SAT. Obviously the graph T is not a k tree; otherwise the thesis would certainly be false. In fact, for instance see [Gol80] pp. 98 100) an algorithm is known that, for any triangulated graph, nds an independent set of vertices of maximum cardinality in polynomial time; by Lemma 4.4, such an algorithm could be used to nd in polynomial time a vertex cover of minimum cardinality for each triangulated graph and then for ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

....= s 0 j#(r 0 ) s 00 j#(r 00 ) s 0 s 00 = s (3) 2) comes from Rule (a) of Denition 3.1. ut 7 5 Adapting the result to other classes of relations The inductive principle underlying the construction of series parallel orders is typical of another class of graphs called cographs [4, 2]. This class, also known as series parallel graphs, may dened by replacing, in Denition 2.2, the ordinal sum by a symmetric series composition. Denition 5.1 Let R = VR ; ER ) and S = VS ; ES ) be two digraphs such that VR VS = We dene the symmetric series composition of R and S to be the ....

Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Academic Press, 1980.

....of NPcompleteness are also discussed in [CLR94] but a more thorough treatment can be found in [GJ79] and [Pap94] 1.1 Graphs There are many textbooks on graph theory. 1 Some of the standard ones are [Har69, BM76, Tut84, CL96] For a focus on algorithmic graph theory, see for example [Eve79, Gol80, GM84, Gib85, Lee90, TS92] and for topological graph theory, see [GT87, BL95] Another recent text is also [Wes96, Wes01] We will now give some definitions and notation concerning graphs that are used throughout the text. A finite, undirected, simple graph G, denoted G = V, E) consists of a ....

Martin C. Golumbic. Algorithmic graph theory and perfect graphs. Academic Press, 1980.

....clique in either the intersection or containment model. Note that many of the definitions used in this chapter come from Golumbic s book, which provides an overview of these graph classes. 2.1 The Intersection Graph Model: The Interval Graph Definition 2. 1 (Interval Intersection Graph) see [Gol80]) A graph G = V; E) is an interval graph if and only if there exists a one to one mapping from the vertices v i of G to intervals c i on the line such that OE I (c i ; c j ) 1 ( v i ; v j ) 2 E. 5 Interval graphs are intersection graphs of intervals on a line. Booth and Lueker [BL76] show ....

....cyclic as in for instance a weekly lecture schedule, placing the intervals on a straight line will not yield the optimal model. It is possible, of course, to place intervals, instead, on a circle, leading to the definition of the circular arc graph class. Definition 2. 2 (Circular arc) see [Gol80]) A graph G = V; E) is a circular arc graph if and only if there exists a one to one mapping from the vertices v i of G to circular arcs c i on a circle such that OE I (c i ; c j ) 1 ( v i ; v j ) 2 E. Every interval graph is a circular arc graph, since the underlying line can be deformed to ....

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press Inc, 1980.

....C visits them in the sequence : a : b : a : b : The interlace graph H = H(C) corresponding to C has the same vertex set as D, with an edge ab in H if a and b are interlaced in C. In the literature, interlace graphs (for example, RR78] are also called circle graphs, as in [Gol80, dF84, GSH89, Spi94]. Trees constitute a particularly tractable set of circle graphs. Definition 2. Given an Euler circuit C with a and b interlaced, a transposition on the pair ab is the circuit C ab resulting from exchanging one of the edge sequences from a to b with the other. See Figure 1 for an example. ....

Martin C. Golumbic, Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.

....the production becomes Kn : Kn 1 that is, to any clique on n elements can be added a new element to create a clique on n 1 elements. 2 Figure 1: A sequence of neighborhood expansions generating chordal graphs clique) and because every chordal graph must have at least two extreme points [8, 6], every chordal graph can be so generated. If the left side of the expansion rule of G chordal is restricted to be only K 1 , i.e. a single point, then G generates all, and only, undirected trees. These two grammars clearly illustrate the essential tree like structure of chordal graphs. This ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

....a directed graph with symmetric pairs of edges. Historically, filled graphs were studied first in the undirected case, specifically for the Cholesky factorization of symmetric positive definite matrices. The theory of undirected filled graphs, which are the same as chordal graphs, is quite rich [19, 30]. We can characterize the structure of G (A) in terms of paths in the graph of A, without actually computing all the elimination graphs. In the following theorem, the paths can be interpreted as directed paths for nonsymmetric matrices and either directed or undirected paths for symmetric ....

....than in the previous section: now A i is always n Theta n, not (n Gamma i) Theta (n Gamma i) 4.1. Nonzero structure of A during elimination. In this subsection we develop a symbolic model of Gaussian elimination with row and or column interchanges. The model is based on that of Golumbic [19] and Gilbert [17] Theorem 4.2 is new. Let H 0 = H(A) be the bipartite graph of A = A 0 . Assume [A 0 ] rc is nonzero and is chosen as pivot at step 1. Define the deficiency of the edge hr 0 ; ci of H 0 to be the set of edges fhi 0 ; ji : c 2 AdjH0 (i 0 ) j 2 AdjH0 (r 0 ) and j = 2 ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

....In Section 5 we define the class of d trapezoid graphs and obtain information about the structure of their separator graphs. Finally, in the last two sections we describe the algorithms for Minimum Fill In and Treewidth of d trapezoid graphs, respectively. 2. Chordal Triangulations We refer to [14] for all graph theoretic notions not defined here. Given a graph G = V; E) we denote by n = jV j its size, by G[W ] the subgraph induced by the vertex set W V , and by G n W the graph G[V n W ] We call any complete subgraph of G a clique. A graph H is called chordal (or triangulated) if it ....

....induced by the vertex set W V , and by G n W the graph G[V n W ] We call any complete subgraph of G a clique. A graph H is called chordal (or triangulated) if it does not contain any induced cycle of length greater than 3. Several other characterizations of chordal graphs are known (cf. [14]) One in terms of minimal separators is given below in Section 3. Every graph is contained in at least one chordal graph with the same vertex set: the complete graph. Any chordal supergraph H G is called a triangulation of MINIMAL SEPARATORS AND CHORDAL TRIANGULATION 3 G. We are looking for ....

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

....which would be sufficient to prove Corollary 3. For the more involved algorithm our basic idea is to use Lov asz s Perfect Graph Theorem, namely that G being perfect is equivalent to (G 0 ) Deltaff(G 0 ) fi fi fiV i G 0 jfi fi fi (5) for all G 0 ind G. See the book of Golumbic [Gol80] for more details and references. Lemma 4. Perfectness can be tested in running time 2 O(n) Algorithm 5 test perfectness Input: Graph G with n vertices Output: true if G is perfect, false otherwise. 1) for all G 0 ind G with 2 vertices do initialize [G 0 ] and ff[G 0 ] 2) for ....

Martin C. Golumbic, Algorithmic graph theory and perfect graphs, Academic Press, London, New York, 1980.

....time O(n log log n) 13] Monotonic subsequences of permutations are of interested in various contexts. It is well known that the upsequences (resp. downsequences) of a permutation are in one to one correspondence with the independent sets (resp. the cliques) of the corresponding permutation graph [5]. Hence, finding the longest upsequence (resp. the longest downsequence) solves the maximum independent set problem (resp. the clique problem) of the corresponding permutation graph. Chang and Wang [3] have reported an O(n log log n) time algorithm for this problem. We are not able to improve they ....

Martin Charles Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

....that for v 1 ; v 2 2 V , w 2 W , fv 1 ; wg; fv 2 ; wg 2 E, the length of the path v 1 w v 2 in G equals the length of the hyperedge of H corresponding to w. b) Special graph classes : Many applications only need solutions for special graph classes such as bipartite graphs, interval graphs [Gol91], planar graphs and grid graphs [Len90, NC88] and many others. In such a case, it may be reasonable not to use a general graph data structure, but to implement a special data structure, which only represents graphs of this class and whose functionality is somewhat different: on one hand, the ....

....as an object of the general graph type, but as a planar map. In this sense the special case hierarchy of graph classes constitutes an isomorphic hierarchy of abstract data types. For instance, grid graphs are a special case of planar graphs, bipartite graphs are a special case of perfect graphs [Gol91], and so on. See [BLS99] for a survey. Moreover, this kind of type specialization can also be used to attack the other parts of Item 1 in Section 3.1. For example, the LEDA class for general undirected graphs is declared to be a specialization of the class for general directed graphs. Hence, an ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1991.

....fixed parameter hierarchy, can be decided in linear time for AT free claw free graphs, for any fixed k. Finally, the minimal triangulations of Pk free graphs, for k 5, are studied by similar methods. 1 Introduction Throughout this paper, we consider simple, finite and undirected graphs. See [14] for all graph theoretic notions not defined here. Three mutually independent vertices of a graph are called an asteroidal triple if, between any two of them, there exists a path that avoids the neighborhood of the third. Graphs without an asteroidal triple are said to be AT free. Lekkerkerker ....

....A minimal separator is called inclusion minimal if it does not contain another minimal separator. A component C of G n S is said to be full if every vertex in S is adjacent to at least one vertex in C. The following crucial property of minimal separators is well known and appears as an exercise in [14]. Lemma 3. Let S be a separator of G and C; D be two components of G n S. Then the following statements are equivalent: 1) C and D are full components of G n S. 2) For every c 2 C and d 2 D, S is a minimal c; d separator. 3) There are some c 2 C and d 2 D such that S is a minimal c; ....

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

....be computed in linear time. The algorithm he proposed can be summarized in four steps: Algorithm 1 Decomposition algorithm Step 1a Build the microstructure of the CSP. Step 2a Triangulate the microstructure [5, 8] Step 3a Compute, in linear time, the maximal cliques of the triangulated graph [6]. Step 4a Determine the subproblemsinduced by the maximal cliques as described in Definition 4. If a maximal clique does not cover all variables, the inducedsubproblem contains no solution to the original CSP and, thus, should be removed. Definition 4 (Jegou) Given a binary CSP P = X;D;C;R) its ....

Martin C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press Inc., New York, New York, 1980.

.... studied by Tucker [10] Finding the chromatic number of these graphs has been proven to be an NP complete problem by Garey et al. 4] In the particular case of proper circulararc graphs, this problem has polynomial complexity [9, 8] Other results concerning circular arc graphs can be found in [5, 6, 7]. Our primary interest in circular arc graph coloring was motivated by the problem of register allocation in loops encountered in programs for high performance microprocessors. In optimizing compilers, register allocation is traditionally performed by coloring (i.e. allocating on registers) the ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

....as well as general properties of these classes. A satisfactory identification of the class corresponding to function 1 Phi 2 (exclusive or) is still open. 1 Introduction Chordal graphs, interval graphs, and indifference graphs have numerous applications and have been extensively studied [Gol80]. Our goal in this paper is to point out that among the several known characterizations for these classes three of them (one for each class) differ only slightly. The difference lies in a certain formula that includes a boolean function of two variables. We then study the classes obtained by using ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

.... Incremental Coloring of Interval Graphs Prashant Pradhan CSE 648 : Advanced Algorithms Project Abstract This work addresses the problem of incrementally coloring interval graphs [1]. The motivation behind the problem comes from finding and maintaining minimum track layouts of a set of DNA segments as segments are added to and deleted from the set. The problem reduces to min coloring of interval graphs. Thus, we seek to efficiently maintain the min coloring of an interval ....

....intervals, with an edge between two vertices iff the segments representing them overlap. Then, the minimum track layout is obtained by min coloring the graph where each color represents a track and all vertices with the same color are assigned to the track represented by that color. As shown in [1], interval graphs belong to the class of triangulated graphs, that can be colored in linear time (O(jV j jEj) Our task is to come up with an incremental algorithm to maintain the min coloring under addition and deletion of intervals. The algorithm to color the graph is not directly extended to ....

Martin Charles Golumbic, "Algorithmic Graph Theory and Perfect Graphs", Academic Press.

....that for v 1 ; v 2 2 V , w 2 W , fv 1 ; wg; fv 2 ; wg 2 E, the length of the path v 1 w v 2 in G equals the length of the hyperedge of H corresponding to w. b) Special graph classes : Many applications only need solutions for special graph classes such as bipartite graphs, interval graphs [Gol91], planar graphs and grid graphs [Len90, NC88] and many others. In such a case, it may be reasonable not to use a general graph data structure, but to implement a special data structure, which only represents graphs of this class and whose functionality is somewhat different: on one hand, the ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1991.

....nodes in the tree are in one one correspondence with cliques of the graph [Ga78,Mo86] Interval graphs are RDV graphs, RDV graphs are UV graphs, and UV graphs are chordal graphs; all the containments are proper [Mo86] To study algorithms on chordal graphs, two bounds are important: Remark 1. 3[Go80]: If G is chordal, then p G nG . Remark 1.4[Go80] If G is chordal then P C2C(G) nC mG nG : Furthermore, the recognition algorithms for chordal graphs cited below can list the vertices in each clique in time O(m n) Interval graphs can be recognized in O(m n) time by an algorithm of Booth and ....

....with cliques of the graph [Ga78,Mo86] Interval graphs are RDV graphs, RDV graphs are UV graphs, and UV graphs are chordal graphs; all the containments are proper [Mo86] To study algorithms on chordal graphs, two bounds are important: Remark 1.3[Go80] If G is chordal, then p G nG . Remark 1. 4[Go80]: If G is chordal then P C2C(G) nC mG nG : Furthermore, the recognition algorithms for chordal graphs cited below can list the vertices in each clique in time O(m n) Interval graphs can be recognized in O(m n) time by an algorithm of Booth and Lueker [Bo76] The recognition algorithm yields ....

Martin Charles Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.

....) 2) s 0 j#(r 0 ) s 00 j#(r 00 ) s 0 s 00 = s (3) 2) comes from Rule (a) of De nition 3.1. ut 5 Adapting the result to other classes of relations The inductive principle underlying the construction of series parallel orders is typical of another class of graphs called cographs [3, 2]. This class, also known as series parallel graphs, may de ned by replacing, in De nition 2.2, the ordinal sum by a symmetric series composition. De nition 5.1 Let R = VR ; ER ) and S = V S ; ES ) be two digraphs such that VR VS = We de ne the symmetric series composition of R and S to be ....

Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Academic Press, 1980.

....running time is essentially linear in the number of dependences. The algorithm results in the minimal number of extra releases (i.e. in excess to those in S 0 ) that are required to synchronize the dependences in D 00 . It does so by computing a minimum cover by cliques on an interval graph [11] representation of the dependence statement intervals. More specifically, consider the dependences in D 00 for which a minimal placement of releases must be found. Based on these dependences, construct an interval graph as follows. For each dependence, create a node in the graph. For simplicity, ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, New York, 1980.

....clockwise) edge of out(z) on C i . We set: I z;e = ae ]L(z) nH ] if z 6= u i 1 or u i 1 = y ]L(z) P i j=0 n j Gamma i] if z = u i 1 and u i 1 6= y (a) b) c) x=u0 y=u3 u1 u2 y=u1 x=u0 u0=x=y=u1 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 (d) x y [2,17] u2 [1,10] [12] 13,17] 1,14] 16,17] i [1,i 1] i 1,17] larger labels smaller labels z Figure 3: Labeling and intervals for the proof of Lemma 3 and I z;e Gamma = ae [1; L(z) if z 6= u i ; if z = u i One can easily check that this labeling and the setting of these intervals build a linear strict ....

....a same connected component (the kernel) with three distinct connected components (the electrons) of exactly two vertices. Note that the vertices of the kernel that connect each electron with the kernel can be distinct of not. It is easy to check that any Y graph is not an interval graph (see [10, 9]) Now, any lithium graph has a Y graph as induced subgraph, and any induced subgraph of an interval graph is an interval graph. Therefore a lithium graph is not an interval graph, that is equivalent to say that any interval graph belongs to 1 LIRS (from Theorem 2) Note that Cn ; n 3 (the ....

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. A Subsidiary of Harcourt Brace Jovanovich, academic press edition, 1980.

.... studied by Tucker [14] Finding the chromatic number of these graphs has been proven to be an NP complete problem by Garey et al. 5] In the particular case of proper circular arc graphs, this problem has polynomial complexity [13, 12] Other results concerning circular arc graphs can be found in [6, 7, 8]. Our primary interest in circular arc graph coloring was motivated by the problem of register allocation in loops encountered in programs for high performance microprocessors. In optimizing compilers, register allocation is traditionally performed by coloring (i.e. allocating on registers) the ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

....C 1 in an increasing order. Repeat this operation considering successively the cycles C i , i = 2; r Gamma 1 until all the vertices of the C i s are labeled. 16] 2 1 0 6 1 1 2 1 0 1 0 3 [1, 0 6 0 [ 3 2 1 1 1 9 4 9 4 14 0 3 14 13 15 12 5 3 2 11 16 10 8 5 4 6 3 2 1 [15,16] [1,13] 12,16] 10,11] 1,8] 2,16] 7 0 5 3 [ u =x=y=u u u y=u u =x u =x (b) r= z) c) y=u u =x (z) L (z) C L (a) z (d) L C C C C e e y=u u Figure 3: Construction of the proof of Lemma 2. Now, we set the intervals as follows (see Figure 3(d) Let nH be the number of vertices of H . ....

....order. Repeat this operation considering successively the cycles C i , i = 2; r Gamma 1 until all the vertices of the C i s are labeled. 16] 2 1 0 6 1 1 2 1 0 1 0 3 [1, 0 6 0 [ 3 2 1 1 1 9 4 9 4 14 0 3 14 13 15 12 5 3 2 11 16 10 8 5 4 6 3 2 1 [15,16] 1,13] [12,16] [10,11] 1,8] 2,16] 7 0 5 3 [ u =x=y=u u u y=u u =x u =x (b) r= z) c) y=u u =x (z) L (z) C L (a) z (d) L C C C C e e y=u u Figure 3: Construction of the proof of Lemma 2. Now, we set the intervals as follows (see Figure 3(d) Let nH be the number of vertices of H . Let n i be the ....

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, Harcourt Brace Jovanovich, Academic Press edition, 1980.

....to find a method of assigning directions to each edge in G c and then show that the resulting orientation is transitive. Proposition 3 The fractional monotone precedence assignment polytope is integral, i. e PF ( N ) P I ( N ) Proof: A transitively orientable graph is a perfect graph [5], hence the constraint graph G c is also a perfect graph. By definition, the fractional node packing polytope of a perfect graph is integral [11] which implies that PF ( N ) is an integral polytope. The above result immediately leads to the integrality of the fractional precedence assignment ....

Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

.... R1,R2,R3 R2,R4 T3 Figure 1: Left: A schedule of seven tasks whose start time and duration has been determined in the scheduling phase. For each task a set of possible resources is shown. Right: the corresponding constraint graph. a special kind of perfect graphs called interval graphs [21, 28]. Although the usual graph coloring problem in interval graphs is known to be linear [22] list coloring is NP complete. 3. Abstraction Abstraction methods have been proposed as promising techniques to reduce the complexity of problem solving and have been applied to a large number of domains . ....

Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press Inc., New York, New York, 1980.

....all the cliques of the constraint graph G c . Therefore, for G c , the rows of the clique matrix are nothing but the rows of M a and M t , and its fractional node packing polytope is the same as its monotone fractional timing assignment polytope, as can be verified from (8) Definition 6: [25]. A graph G = fV; Eg is called transitively orientable if each edge can be assigned a one way direction in such a way that the resulting oriented graph (V; F ) satisfies the following property: a; b) 2 F and (b; c) 2 F implies (a; c) 2 F Proposition 2: The constraint graph G c is transitively ....

....6. It is obvious that G s satisfies the transitive property, because G s itself is the transitive closure of digraph G s . Proposition 3: The fractional monotone timing assignment polytope is integral, i. e PF ( N ) P I ( N ) Proof: A transitively orientable graph is a perfect graph [25], hence the constraint graph G c is also a perfect graph. By definition, the fractional node packing polytope of a perfect graph is integral [10] which implies that PF ( N ) is an integral polytope. The above result immediately leads to the integrality of the fractional timing assignment ....

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Martin Charles Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.

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Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

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Martin C. Golumbic. Algorithmic graph theory and perfect graphs. Academic Press, 1980.

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Martin Ch. Golumbic. Algorithmic Graph Theory And Perfect Graphs. Academic Press, New York, 1980.

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Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

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Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.

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Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press Inc., New York, New York, 1980.

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Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press Inc., New York, NY, 1980.

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