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On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four–Extended Abstract–
On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four – Extended Abstract – Arnaud Pˆecher ½ Laboratoire Bordelais de Recherche Informatique (LaBRI) 351 cours de la Lib´eration, 33405 Talence, France [email protected]
Annegret K. Wagler ¾ Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin (ZIB) Takustr. 7, 14195 Berlin, Germany [email protected]
Abstract Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs have rank facets only [3,12] while there are examples with clique with clique number having non-rank facets [8,10]. The aim of the present paper is to treat the number remaining case with clique number : we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets. Key words: web, rank-perfect graph, stable set polytope, (non-)rank facet
A natural generalization of odd holes and odd antiholes are graphs with circular symmetry of their maximum cliques and stable sets, called webs: a web is a where is an edge if and differ by at most and graph with nodes .
½ ¾
this work was supported by DONET/ZIB. this work supported by the Deutsche Forschungsgemeinschaft (Gr 883/9–1).
Preprint submitted to Elsevier Science
30 January 2003
The webs with on nine nodes are depicted in Figure 1. Notice that with webs are also called circulant graphs [2]. Furthermore, graphs , and were introduced in [12].
1
W 92
W9
W 39
Figure 1 Webs and line graphs belong to the classes of quasi-line graphs (the neighborhood of any node can be partitioned into two cliques) and claw-free graphs (the neighborhood of any node does not contain a stable set of size 3). The line graph of a graph is obtained by taking the edges of as nodes of and connecting two nodes in iff the corresponding edges of are incident.
Webs and line graphs are relevant w.r.t. describing the stable set polytopes of those larger graph classes [5,6,10]. The stable set polytope STAB of is defined as . the convex hull of the incidence vectors of all stable sets of the graph ¼ A rank constraint associated with an induced subgraph is the 0/1-constraint of the form ¼
¾¼
(where ¼ denotes the cardinality of a maximum stable set in ¼ ). A graph is rank-perfect if all non-trivial facets of its stable set polytope are rank constraints. The class of rank-perfect graphs contains all perfect graphs due to [1] since all non-trivial facets of their stable set polytopes are rank constraints ¾
associated with (inclusion-wise maximal) cliques is a clique).
(note ¼ holds iff ¼
All facets of the stable set polytope of line graphs are known from matching theory [4]. In particular, line graphs are rank-perfect. A characterization of the rank facets in stable set polytope of claw-free graphs was given by G ALLUCCIO & S AS SANO [5]. They showed that all rank facets can be constructed by means of standard operations from rank constraints associated with cliques, partitionable webs, or line graphs of 2-connected, critical hypomatchable graphs. However, we are still far from having a complete description for the stable set polytopes of webs and, therefore, of quasi-line and claw-free graphs, too. Finding a decent linear description of the stable set polytopes of claw-free graphs is a 2
long-standing problem [7]. Claw-free graphs are not rank-perfect: G ILES & T ROTTER [6], O RIOLO [10], and L IEBLING ET AL [9] found non-rank facets which occur even in the stable set polytopes of quasi-line graphs. These non-rank facets rely on combinations of joined webs. Several further authors studied the stable set polytopes of webs. Obviously, webs with clique number 2 are holes, hence they are perfect if is even and minimally imperfect if is odd. Thus, all webs with clique number 2 are rank-perfect due to [1,11]. DAHL [3] studied webs with clique number 3 and showed that they are rank-perfect. On the other hand, K IND [8] found (by means of the PORTA software ) examples of webs with clique number the stable set polytopes of , , , , . O RIOLO [10] and which have non-rank facets, e.g., , L IEBLING ET AL [9] presented further examples of such webs and asked whether the stable set polytopes of webs with clique number admit rank facets only.
The aim of the present paper is to answer that question by providing an infinite sequence of webs with clique number the stable set polytopes of which have non-rank facets. For that, we first analyze the structure of the known non-rank facets of webs with higher clique number. Then we investigate a similar construction for the webs that gives rise to an infinite sequence of webs with clique number having nonrank facets: Theorem 1 Let then
with . Let be an induced subweb of
¾¾¾
is a non-rank facet of STAB(
¾¾¾
).
Theorem 1 provides an infinite sequence of not rank-perfect webs with clique , , , , ... answering the question whether the webs number 4, namely with clique number 4 are rank-perfect or not.
References
[1] V. Chv´atal, On Certain Polytopes Associated with Graphs. J. Combin. Theory (B) 18 (1975) 138–154 [2] V. Chv´atal, On the Strong Perfect Graph Conjecture. J. Combin. Theory (B) 20 (1976) 139–141
By PORTA it is possible to generate all facets of the convex hull of a given set of integer points, see http://www.zib.de
3
[3] G. Dahl, Stable Set Polytopes for a Class of Circulant Graphs. SIAM J. Optim. 9 (1999) 493–503 [4] J.R. Edmonds and W.R. Pulleyblank, Facets of 1-Matching Polyhedra. In: C. Berge and D.R. Chuadhuri (eds.) Hypergraph Seminar. Springer-Verlag, Heidelberg, (1974), 214–242 [5] A. Galluccio and A. Sassano, The Rank Facets of the Stable Set Polytope for Claw-Free Graphs. J. Comb. Theory B 69 (1997) 1–38 [6] R. Giles and L.E. Trotter, jr., On Stable Set Polyhedra for J. Comb. Theory B 31 (1981) 313–326
½ ¿-free
Graphs.
[7] M. Gr¨otschel, L. Lov´asz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, (1988) [8] J. Kind, Mobilit¨atsmodelle f¨ur zellulare Mobilfunknetze: Produktformen und Blockierung. PhD thesis, RWTH Aachen (2000) [9] T.M. Liebling, G. Oriolo, B. Spille, and G. Stauffer, On Non-Rank Facets of the Stable Set Polytope of Claw-Free Graphs and Circulant Graphs. Submitted to SIAM J. on Discrete Math. [10] G. Oriolo, Clique Family Inequalities for the Stable Set Polytope for Quasi-Line Graphs. To appear in: Special Issue of Distcrete Applied Math. on Stability Problems, V. Lozin and D. de Werra (eds.) [11] M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180–196 [12] L.E. Trotter, jr., A Class of Facet Producing Graphs for Vertex Packing Polyhedra. Discrete Math. 12 (1975) 373–388
4
Annegret K. Wagler ¾ Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin (ZIB) Takustr. 7, 14195 Berlin, Germany [email protected]
Abstract Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs have rank facets only [3,12] while there are examples with clique with clique number having non-rank facets [8,10]. The aim of the present paper is to treat the number remaining case with clique number : we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets. Key words: web, rank-perfect graph, stable set polytope, (non-)rank facet
A natural generalization of odd holes and odd antiholes are graphs with circular symmetry of their maximum cliques and stable sets, called webs: a web is a where is an edge if and differ by at most and graph with nodes .
½ ¾
this work was supported by DONET/ZIB. this work supported by the Deutsche Forschungsgemeinschaft (Gr 883/9–1).
Preprint submitted to Elsevier Science
30 January 2003
The webs with on nine nodes are depicted in Figure 1. Notice that with webs are also called circulant graphs [2]. Furthermore, graphs , and were introduced in [12].
1
W 92
W9
W 39
Figure 1 Webs and line graphs belong to the classes of quasi-line graphs (the neighborhood of any node can be partitioned into two cliques) and claw-free graphs (the neighborhood of any node does not contain a stable set of size 3). The line graph of a graph is obtained by taking the edges of as nodes of and connecting two nodes in iff the corresponding edges of are incident.
Webs and line graphs are relevant w.r.t. describing the stable set polytopes of those larger graph classes [5,6,10]. The stable set polytope STAB of is defined as . the convex hull of the incidence vectors of all stable sets of the graph ¼ A rank constraint associated with an induced subgraph is the 0/1-constraint of the form ¼
¾¼
(where ¼ denotes the cardinality of a maximum stable set in ¼ ). A graph is rank-perfect if all non-trivial facets of its stable set polytope are rank constraints. The class of rank-perfect graphs contains all perfect graphs due to [1] since all non-trivial facets of their stable set polytopes are rank constraints ¾
associated with (inclusion-wise maximal) cliques is a clique).
(note ¼ holds iff ¼
All facets of the stable set polytope of line graphs are known from matching theory [4]. In particular, line graphs are rank-perfect. A characterization of the rank facets in stable set polytope of claw-free graphs was given by G ALLUCCIO & S AS SANO [5]. They showed that all rank facets can be constructed by means of standard operations from rank constraints associated with cliques, partitionable webs, or line graphs of 2-connected, critical hypomatchable graphs. However, we are still far from having a complete description for the stable set polytopes of webs and, therefore, of quasi-line and claw-free graphs, too. Finding a decent linear description of the stable set polytopes of claw-free graphs is a 2
long-standing problem [7]. Claw-free graphs are not rank-perfect: G ILES & T ROTTER [6], O RIOLO [10], and L IEBLING ET AL [9] found non-rank facets which occur even in the stable set polytopes of quasi-line graphs. These non-rank facets rely on combinations of joined webs. Several further authors studied the stable set polytopes of webs. Obviously, webs with clique number 2 are holes, hence they are perfect if is even and minimally imperfect if is odd. Thus, all webs with clique number 2 are rank-perfect due to [1,11]. DAHL [3] studied webs with clique number 3 and showed that they are rank-perfect. On the other hand, K IND [8] found (by means of the PORTA software ) examples of webs with clique number the stable set polytopes of , , , , . O RIOLO [10] and which have non-rank facets, e.g., , L IEBLING ET AL [9] presented further examples of such webs and asked whether the stable set polytopes of webs with clique number admit rank facets only.
The aim of the present paper is to answer that question by providing an infinite sequence of webs with clique number the stable set polytopes of which have non-rank facets. For that, we first analyze the structure of the known non-rank facets of webs with higher clique number. Then we investigate a similar construction for the webs that gives rise to an infinite sequence of webs with clique number having nonrank facets: Theorem 1 Let then
with . Let be an induced subweb of
¾¾¾
is a non-rank facet of STAB(
¾¾¾
).
Theorem 1 provides an infinite sequence of not rank-perfect webs with clique , , , , ... answering the question whether the webs number 4, namely with clique number 4 are rank-perfect or not.
References
[1] V. Chv´atal, On Certain Polytopes Associated with Graphs. J. Combin. Theory (B) 18 (1975) 138–154 [2] V. Chv´atal, On the Strong Perfect Graph Conjecture. J. Combin. Theory (B) 20 (1976) 139–141
By PORTA it is possible to generate all facets of the convex hull of a given set of integer points, see http://www.zib.de
3
[3] G. Dahl, Stable Set Polytopes for a Class of Circulant Graphs. SIAM J. Optim. 9 (1999) 493–503 [4] J.R. Edmonds and W.R. Pulleyblank, Facets of 1-Matching Polyhedra. In: C. Berge and D.R. Chuadhuri (eds.) Hypergraph Seminar. Springer-Verlag, Heidelberg, (1974), 214–242 [5] A. Galluccio and A. Sassano, The Rank Facets of the Stable Set Polytope for Claw-Free Graphs. J. Comb. Theory B 69 (1997) 1–38 [6] R. Giles and L.E. Trotter, jr., On Stable Set Polyhedra for J. Comb. Theory B 31 (1981) 313–326
½ ¿-free
Graphs.
[7] M. Gr¨otschel, L. Lov´asz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, (1988) [8] J. Kind, Mobilit¨atsmodelle f¨ur zellulare Mobilfunknetze: Produktformen und Blockierung. PhD thesis, RWTH Aachen (2000) [9] T.M. Liebling, G. Oriolo, B. Spille, and G. Stauffer, On Non-Rank Facets of the Stable Set Polytope of Claw-Free Graphs and Circulant Graphs. Submitted to SIAM J. on Discrete Math. [10] G. Oriolo, Clique Family Inequalities for the Stable Set Polytope for Quasi-Line Graphs. To appear in: Special Issue of Distcrete Applied Math. on Stability Problems, V. Lozin and D. de Werra (eds.) [11] M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180–196 [12] L.E. Trotter, jr., A Class of Facet Producing Graphs for Vertex Packing Polyhedra. Discrete Math. 12 (1975) 373–388
4