V. Chvatal, On certain polytopes associated with graphs. J. Combin. Theory Ser. B 18: 138-154, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a graph, due to the fact that we can optimize any linear function over any of N (G) in polynomial time, provided k = O(1) investigating N 0 , N and N ranks of graphs and further understanding of graphs of small rank remain very interesting. 3 An Elementary Decomposition Chv atal [4] has shown that if the graph G can be decomposed into two parts, G 1 and G 2 , so that their intersection is a complete graph, then the facets of STAB(G) are just the union of the facets of STAB(G 1 ) and STAB(G 2 ) Hence we get a similar property for the N 0 rank, N rank and the N rank: ....

....implies that G k 1 is critical, thus so is G k 1 , and it is easy to see that (G k ) k. Notice that we can have the edge v 3j 4 v 3j instead of the edge v 3j 4 v 3j 2 for any 2 j k, and the conclusion of Lemma 36 would still apply, so the resulting graph is also critical. Chv atal [4] proved the following nice property of critical graphs: Theorem 38 If the graph G is critical, then e x (G) de nes a facet of STAB(G) We can now establish the N 0 rank of the graphs G k : Theorem 39 r 0 (G k ) blog 2 3 c 2 for any k 1. Proof: Since by Lemma 36 the graph G k ....

V. Chvatal, On certain polytopes associated with graphs, J. of Combin. Theory Ser. B 18 (1975) 138-154.

....have at most one node in common) The stable set polytope of every graph is contained in the polytope given by the nonnegativity constraints and clique constraints. Perfect graphs are precisely those graphs for which nonnegativity and clique constraints suce to describe their stable set polytope [2]. Following a suggestion of Gr otschel, 2 Lov asz, and Schrijver [5] one may relax the notion of perfectness by generalizing clique constraints to other classes of inequalities valid for the stable set polytope and then by investigating all graphs such that their stable set polytope is ....

....relations of these polytopes: STAB(G) WSTAB(G) RSTAB(G) QSTAB(G) 3 Each of the polytopes QSTAB(G) RSTAB(G) WSTAB(G) is a relaxation of STAB(G) Very often, these inclusions are proper. In a few important cases, however, equality holds. Most notably, perfect graphs are characterized in [2] to be precisely the graphs G with QSTAB(G) STAB(G) Let us call all graphs G with RSTAB(G) STAB(G) rank perfect. Every perfect graph is obviously an example of a rank perfect graph. Padberg [11] showed that the stable set polytope of minimally imperfect graphs G admits, besides facets of ....

V. Chvatal, On Certain Polytopes Associated with Graphs. J. Combin. Theory (B) 18 (1975) 138-154

.... n nodes that has diameter one for all n 2 (Barahona and Mahjoub [6] Other polytopes turned out to have more complicated graphs, e.g. the stable set polytopes, for which two vertices are adjacent if and only if the symmetric di erence of the corresponding stable sets induces a connected graph [8]. Another interesting example is the basis polytope of a matroid (i.e. the convex hull of the characteristic vectors of its bases) where two vertices are adjacent if and only if the corresponding bases have a symmetric di erence of cardinality two (observed by Edmonds in the early 1970 s) All ....

.... of two edges (since the spanning trees of some graph are the bases of a matroid, the graphic matroid de ned by that graph) As mentioned above, two vertices of a stable set polytope are adjacent if and only if the symmetric di erence of the corresponding stable sets induces a connected subgraph [8]. Since matchings correspond to stable sets in the line graph, two vertices of a matching polytope thus are adjacent if and only if the symmetric di erence of the corresponding matchings is connected. The same is true for perfect matching polytopes, since they are faces of matching polytopes. Two ....

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V. Chvatal. On certain polytopes associated with graphs. J. Comb. Theory, Ser. B, 18:138-154, 1975.

....notion of perfect graph. A graph G is perfect if, for every induced subgraph H of G, the chromatic number of H equals the size of its largest clique. The fundamental connection between the theory of perfect graphs and integer programming was established by Fulkerson [33] Lov asz [48] and Chv atal [13]. The clique node matrix of a graph G is a 0; 1 matrix whose columns are indexed by the nodes of G and whose rows are the incidence vectors of the maximal cliques of G. Theorem 6.1 (Lov asz [48] Fulkerson [33] Chv atal [13] Let A be a 0,1 matrix. The set packing polytope P (A) is integral if ....

....programming was established by Fulkerson [33] Lov asz [48] and Chv atal [13] The clique node matrix of a graph G is a 0; 1 matrix whose columns are indexed by the nodes of G and whose rows are the incidence vectors of the maximal cliques of G. Theorem 6. 1 (Lov asz [48] Fulkerson [33] Chv atal [13]) Let A be a 0,1 matrix. The set packing polytope P (A) is integral if and only if the rows of A of maximal support form the clique node matrix of a perfect graph. Now we extend the de nition of perfect and ideal 0; 1 matrices to 0; 1 matrices. A 0; 1 matrix A is ideal if the generalized set ....

V. Chvatal, On certain polytopes associated with graphs, Journal of Combinatorial Theory B 18 (1975) 138-154.

....1. A graph is perfect if the node chromatic number is equal to the maximum cardinality of a clique in every node induced subgraph of the given graph. The following well known Proposition explains the connection between 0; 1 perfect matrices and perfect graphs (see [10] for a survey) Theorem 1 ([4, 6, 9]) A 0; 1 matrix A is perfect if and only if the following two conditions hold: 1. the graph GA is perfect, 2. the incidence vector of any maximal clique of graph GA is a row of matrix A. Therefore testing whether a given 0; 1 matrix A is perfect is equivalent to testing if both conditions (1) ....

Chv'atal, V.: On certain polytopes associated with graphs. J. Comb. Theory, Ser. B, 18, 138-154 (1975).

....of pairs of corresponding vertices (v; 1) v; 2) of the two copies of G. Furthermore, the constraints (2) correspond to the cliques in each of the copies of G. Thus, the polyhedron P (M) has only integral extreme points (or equivalent M is perfect) if and only if G K 2 is perfect (Chv atal [4]) Therefore, the goal is to nd a characterization of graphs G such that G K 2 is perfect. We note that G K 2 is a bipartite graph, if G is bipartite; see also Figure 2. The solution x found by the linear program may not correspond directly to a solution of the coloring problem. If k v;3 ....

....for recognition of parity graphs are presented in [1, 12] 4 Main theorem In this section, we prove the following result: Theorem 4.1 The following two statements are equivalent: 1) G K 2 is a perfect graph. 2) G is a parity graph. Using this characterization and the theorem of Chv atal [4] (see also the section about perfect matrices) we obtain directly: 5 Corollary 4.1 Let I be an instance of the GOCCP problem containing a graph G = V; E) with n vertices and coloring costs k v;c such that k v;c = k v;3 for c 3 and v 2 V . Then, we have the following equivalence: The ....

V. Chvatal, On certain polytopes associated with graphs, Journal Combinatorial Theory B-18 (1975), 138-154.

....for constructing a weakened set packing relaxation in this way is that this makes separation easier. The main property is that most violated cycle, clique, and orthonormal representation constraints have a very restricted support. Namely, denote for each 4 G 1 2 3 4 5 6 7 8 9 e G [1] 2] 5] [7] [9] 1 2 3 4 5 6 7 8 9 G= 1] 2] 5] 7] 9] 1] 2] 5] 7] 9] Figure 2: Constructing a Class Con ict Set Packing Relaxation. x 0 2 R n 0 and every equivalence class [v] by [v] x 0 : argmax x 0 [v] a representative from [v] of maximum x 0 value, breaking ties, say, by index, and ....

....relaxation in this way is that this makes separation easier. The main property is that most violated cycle, clique, and orthonormal representation constraints have a very restricted support. Namely, denote for each 4 G 1 2 3 4 5 6 7 8 9 e G [1] 2] 5] 7] 9] 1 2 3 4 5 6 7 8 9 G= 1] 2] 5] [7] [9] 1] 2] 5] 7] 9] Figure 2: Constructing a Class Con ict Set Packing Relaxation. x 0 2 R n 0 and every equivalence class [v] by [v] x 0 : argmax x 0 [v] a representative from [v] of maximum x 0 value, breaking ties, say, by index, and collect these nodes in the set V x 0 : ....

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Chvatal (1975). On Certain Polytopes Associated with Graphs. J. Comb. Theory 18, 138-154.

....of (SSP) STAB(G) satisfies the nonnegativity inequalities x i 0 (i 2 V ) and the clique inequalities X i2C x i 1 (C : clique) Let us denote by QSTAB(G) the polytope with the nonnegativity inequalities and the clique inequalities. Then we have STAB(G) QSTAB(G) Moreover Theorem 2. 2 ([9, 4]) STAB(G) QSTAB(G) if and only if G is a perfect graph. For definition and results of perfect graphs including Theorem 2.2, see [10] Theorem 2.2 is a polyhedral characterization of perfect graphs. Note that it is still NP hard to optimize w T x over QSTAB(G) 11] Grotschel, Lov asz, and ....

V. Chv' atal, On certain polytopes associated with graphs, J. Combin. Theory Ser. B, 18 (1975), pp. 138--154.

....pairs of corresponding vertices (v; 1) v; 2) of the two copies of G. Furthermore, the constraints (2) correspond to the cliques in each of the copies of G. Thus, the polyhedron P (M) has only integral extreme points (or equivalent M is perfect) if and only if G Theta K 2 is perfect (Chv atal [4]) Therefore, the goal is to find a characterization of graphs G such that G Theta K 2 is perfect. We note that G Theta K 2 is a bipartite graph, if G is bipartite; see also Figure 2. The solution x found by the linear program may not correspond directly to a solution of the coloring problem. If ....

....9 Figure 3: A partial list of forbidden subgraphs. 4 Main theorem In this section, we prove the following result: Theorem 4.1 The following two statements are equivalent: 1) G Theta K 2 is a perfect graph. 2) G is a parity graph. Using this characterization and the theorem of Chv atal [4] (see also the section about perfect matrices) we obtain directly: Corollary 4.1 Let I be an instance of the GOCCP problem containing a graph G = V; E) with n vertices and coloring costs k v;c such that k v;c = k v;3 for c 3 and v 2 V . Then, we have the following equivalence: The polyhedron P ....

V. Chv'atal, On certain polytopes associated with graphs, Journal Combinatorial Theory B-18 (1975), 138--154.

....an even subdivision of K 4 (a subdivision with an even number of new nodes on each edge) Sur anyi [12] classified ff critical graphs with ffi = 3. Lov asz [7] proved that ff critical graphs with a fixed defect can be obtained from a finite number of basic graphs by even subdivision. Chv atal [3] established a rather interesting connection between ff critical graphs and polyhedral combinatorics by showing that if G is a connected ff critical graph, then the inequality P v2V x v ff(G) defines a facet of the stable set polytope of G. Thus every facet of the stable set polytope can be ....

V. Chv'atal, On certain polytopes associated with graphs, J. of Combinatorial Theory (B) 18 (1975), 138--154.

....or the stable set problem. Let X V P G be the set of incidence vectors corresponding to feasible vertex packings in the graph G, and let ff(G) be the maximum cardinality of a vertex packing in G. An edge is called critical if its removal from G produces a graph G 0 with ff(G 0 ) ff(G) Chv atal (1975) derived the following general sufficient condition for an inequality to define a facet of conv(X VP G) Theorem 1 Chv atal (1975) Let E be the set of critical edges of G. If the graph G = V; E ) is connected, then the inequality P j2V x j ff(G) defines a facet of conv (X V P G) ....

....and let ff(G) be the maximum cardinality of a vertex packing in G. An edge is called critical if its removal from G produces a graph G 0 with ff(G 0 ) ff(G) Chv atal (1975) derived the following general sufficient condition for an inequality to define a facet of conv(X VP G) Theorem 1 Chv atal (1975). Let E be the set of critical edges of G. If the graph G = V; E ) is connected, then the inequality P j2V x j ff(G) defines a facet of conv (X V P G) A clique in a graph G is a subgraph of G where each two vertices are connected by an edge, see Figure 1a. A clique is maximal if it ....

V. Chv' atal (1975) "On certain polytopes associated with graphs" Journal of Combinatorial Theory B 18 138--154.

....taking the floor of the right hand side, we obtain the following weak odd closed walk inequality k X i=1 x i Gamma X i2F x i k Gamma 1 2 ; 5) that is valid for all integral points x 2 P . Well known examples of such inequalities are the odd hole inequalities of the stable set polytope [Chv75], and the weak odd closed alternating trail inequalities of the asymmetric traveling salesman polytope [Bal89] see also [CF95] Examples in which the original matrix has also entries equal to Gamma1 are 2 chorded odd cycle inequalities of the clique partitioning polytope [GW90] odd wheel ....

V. Chv'atal. On certain polytopes associated with graphs. Journal of Combinatorial Theory Ser. B, 13:138 -- 154, 1975.

....just two vertices. Papers that studied stable set Research partially supported by scholarships from the Ontario Ministry of Colleges and Universities. y Research partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. polytopes include Chv atal [9], Fonlupt and Uhry [11] Gerards [12] Giles and Trotter [13] Mahjoub [16] Nemhauser and Trotter [17] Padberg [19] Tesch [20] Trotter [21] and Wolsey [22] see also Grotschel, Lov asz and Schrijver [14] In this paper, we introduce a large class of valid inequalities, which we call wheel ....

....7: Theorem 2.1 Let G be a graph and W be a simple 1 wheel that is a subgraph of G. Then the inequalities I W E (that is, I W A ) and I W O (that is, I W B ) are both valid for PG . 2 Instances of 1 wheel inequalities have occurred repeatedly in the literature; see, for example, Chv atal [9], Grotschel, Lov asz and Schrijver [14] and Barahona and Mahjoub [1] Although up to now no general classes seem to have been defined, 14] page 301, does refer (without definition) to such a class. For k = 1, the 1 wheel reduces to an odd K 4 which is an important structure in the study of ....

V. Chv'atal, "On certain polytopes associated with graphs", Journal of Combinatorial Theory, Series B 18, 138--154 (1975).

....n maximum stable set incidence vectors, and these vectors are linearly independent. Finally, note that if W n ff Gamma1 is a prime web, then the edges of H (as defined above) are ff critical in W n ff Gamma1 and form a spanning connected subgraph. Thus W n ff Gamma1 induces a rank facet (see [4]) 2 An important class of graphs in the study of perfect graphs are the so called partitionable graphs. G is partitionable if it has ff 1 nodes and for each node v, G Gamma v has an colouring and an ff clique cover. Results in [2] show that each partitionable graph is patterned and hence ....

V. Chv'atal, On certain polytopes associated with graphs, Journal of Combinatorial Theory B, 18, (1975), 138-154.

....Using Branch and Cut 8 the edge node incidence matrix of GC , then the node packing polytope P C = convfx 2 B n : AC x 5 1g satisfies P P C . Consequently, P IP P C , and one can generate valid inequalities for P IP by applying results from the theory of node packing polytopes (Chv atal 1975, Padberg 1973, Trotter 1975, Wolsey 1976b) The following example illustrates that cuts generated via the conflict graph are, in general, tighter than cuts obtained using only one constraint at a time: Example 1. Let P IP = convfx 2 B 5 : Ax = bg, where A = 2 6 4 1 1 1 0 0 0 1 1 0 1 1 1 0 2 ....

Chv' atal, V. 1975. On Certain Polytopes Associated with Graphs. Journal of Combinatorial Theory B13, 138--154.

....packing polytopes to equality constrained 0 1 polytopes by introducing 0 1 valued slack variables. In the paper [1] Balas and Padberg discussed the adjacency structures of set partitioning polytopes. The adjacency criterion of perfect matching polytopes was derived by Balinski [5] and Chv atal [9]. Ikura and Nemhauser discussed the adjacency structures of set packing polytopes in [16] and extended some properties of set partitioning polytopes showed by Balas and Padberg in [1, 2, 3] The adjacency criterion of vertex packing polytopes was discussed by Trubin [25] and Chv atal [9] Section ....

.... Chv atal [9] Ikura and Nemhauser discussed the adjacency structures of set packing polytopes in [16] and extended some properties of set partitioning polytopes showed by Balas and Padberg in [1, 2, 3] The adjacency criterion of vertex packing polytopes was discussed by Trubin [25] and Chv atal [9]. Section 2 of this paper considers the equality constrained 0 1 polytopes. In particular, we establish two fundamental properties of the equality constrained 0 1 polytopes, which are useful to discuss the adjacency structures of some combinatorial polyhedra. In Sections 3,4, we consider the ....

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V. Chv'atal, On certain polytopes associated with graphs, J. Combin. Theory, Ser. B 18 (1975) 138--154.

....of constraints equal to N. On the other hand this set of constraints may be relative easier to deal with in comparison to the others. It can be explained by the capacitated facility location formulation we give to the problem. We may see constraints (2) as a facet of the vertex packing polytope [3] in a bipartite graph, which lead to the linearization of the x ij variables in the model. The set of constraints (5) and (6) however, may not present such a simple structure as it may be seen as graph coloring like constraints [10] So given the model dimension and its structure we decompose ....

V. Chvátal, On Certain Polytopes Associated with Graphs, Journal of Combinatorial Theory B18 (1975).

....showed that the problem of checking non adjacency on the travelling salesman polytope is NP complete. So, one cannot expect an efficient edge following type algorithm for the travelling salesman problem. However, there exist some classes of combinatorial polytopes, including matching polytopes [4, 6], vertex packing polytopes [17, 6] set partitioning polytopes [1, 2, 3] and set packing polytopes [12] such that we can decide the adjacency of two given vertices in polynomial time. In this paper, we show that for some well known classes of combinatorial polytopes, the non adjacency test ....

....non adjacency on the travelling salesman polytope is NP complete. So, one cannot expect an efficient edge following type algorithm for the travelling salesman problem. However, there exist some classes of combinatorial polytopes, including matching polytopes [4, 6] vertex packing polytopes [17, 6], set partitioning polytopes [1, 2, 3] and set packing polytopes [12] such that we can decide the adjacency of two given vertices in polynomial time. In this paper, we show that for some well known classes of combinatorial polytopes, the non adjacency test problems are NP complete. We deal with ....

[Article contains additional citation context not shown here]

V. Chv'atal, On certain polytopes associated with graphs, J. Combin. Theory, Ser. B 18 (1975) 138--154.

....taking the floor of the right hand side, we obtain the following weak odd closed walk inequality k X i=1 x i Gamma X i2F x i k Gamma 1 2 ; 5) that is valid for all integral points x 2 P . Well known examples of such inequalities are the odd hole inequalities of the stable set polytope [Chv75], and the weak odd closed alternating trail inequalities of the asymmetric TSP polytope [Bal89] see also [CF93] Examples in which the original matrix contains coefficients equal to Gamma1 are 2 chorded odd cycle inequalities of the clique partitioning polytope [GW90] and odd cycle ....

V. Chv'atal. On certain polytopes associated with graphs. Journal of Combinatorial Theory Ser. B, 13:138--154, 1975.

....taking the floor of the right hand side, we obtain the following weak odd closed walk inequality k X i=1 x i Gamma X i2F x i k Gamma 1 2 ; 5) that is valid for all integral points x 2 P . Well known examples of such inequalities are the odd hole inequalities of the stable set polytope [Chv75], and the weak odd closed alternating trail inequalities of the asymmetric TSP polytope [Bal89] see also [CF95] Examples in which the original matrix has entries equal to Gamma1 are 2 chorded odd cycle inequalities of the clique partitioning polytope [GW90] odd wheel inequalities of the graph ....

V. Chv'atal. On certain polytopes associated with graphs. Journal of Combinatorial Theory Ser. B, 13:138 -- 154, 1975.

....or the stable set problem. Let X V P G be the set of incidence vectors corresponding to feasible vertex packings in the graph G, and let ff(G) be the maximum cardinality of a vertex packing in G. An edge is called critical if its removal from G produces a graph G 0 with ff(G 0 ) ff(G) Chv atal (1975) derived the following general sufficient condition for an inequality to define a facet of conv(X VP G) Theorem 1 Chv atal (1975) Let E be the set of critical edges of G. If the graph G = V; E ) is connected, then the inequality P j2V x j ff(G) defines a facet of conv (X V P G) ....

....and let ff(G) be the maximum cardinality of a vertex packing in G. An edge is called critical if its removal from G produces a graph G 0 with ff(G 0 ) ff(G) Chv atal (1975) derived the following general sufficient condition for an inequality to define a facet of conv(X VP G) Theorem 1 Chv atal (1975). Let E be the set of critical edges of G. If the graph G = V; E ) is connected, then the inequality P j2V x j ff(G) defines a facet of conv (X V P G) A clique in a graph G is a subgraph of G where each two vertices are connected by an edge, see Figure 1a. Since no two vertices in ....

V. Chv' atal (1975) "On certain polytopes associated with graphs" Journal of Combinatorial Theory B 18 138--154.

....1 2 jE 0 j ; 14) where H ae V and where the edges in E 0 have precisely one endvertex in H. Note that only 2 matching constraints with an odd number of edges in E 0 can be facet defining, since they are otherwise implied by the degree constraints. Comb inequalities were introduced by Chv atal (1975) as a generalization of the 2 matching constraints. In the comb inequalities the edges in E 0 are replaced by an odd number, s, of disjoint vertex sets T 1 ; T s , called teeth, each having one vertex in common with the handle H. The comb inequality is written as x(E(H) s X j=1 ....

V. Chv' atal (1975) "On certain polytopes associated with graphs" Journal of Combinatorial Theory B 18 138--154.

.... constitutes the unique minimal defining linear system of STAB(G) The basic properties of the polytope STAB(G) and the crucial connections with the theory of Perfect Graphs have been studied by several authors (see [10] for a survey) following the fundamental papers due to Padberg [15] Chvatal [4] and Nemhauser and Trotter [14] In particular, the above papers considered with a special attention a fundamental class of valid inequalities for STAB(G) the rank inequalities. The rank inequality associated with a subset T of V is an inequality of the form P v#T x v # #(T ) Clearly, a ....

....to provide a satisfactory characterization of the graphs G whose associated rank inequalities define facets for STAB(G) and R(G) This task appears indeed very di#cult, nevertheless, some interesting necessary or su#cient conditions have been proposed in the literature. In particular, Chvatal [4] called # critical the edge e # E with the property that #(G e) #(G) 1 and proved that if the graph G # = V, E # ) with E # = e # E : e is # critical ) is connected, then the rank inequality P v#V x v # #(G) defines a facet of STAB(G) Conversely, Balas and Zemel [2] proved ....

[Article contains additional citation context not shown here]

V.CHV ATAL, On certain polytopes associated with graphs, J. Combinatorial Theory Ser. B, 18, 138-154 (1975).

....rows of A are the characteristic vectors of cliques, and the vertices of P are the characteristic vectors of stable sets of a perfect graph. This is a straightforward consequence of Lov asz s Perfect Graph Theorem (1972 a) and Fulkerson s theorem on antiblocking (1970,1971) as it was observed by Chv atal (1975). Thus Perfect Graphs provide a pleasant reformulation of a wide range of integer programs with advantageous properties with respect to Optimization. While the Perfect Graph Theorem has become the material of undergraduate textbooks, the interest has decreased toward the structure of perfect or ....

V. Chv'atal (1975) "On certain polytopes associated with graphs", Journal of Combinatorial Theory/B, 18, 138--154.

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V. Chvatal, On certain polytopes associated with graphs. J. Combin. Theory Ser. B 18: 138-154, 1975.

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