- Email: [email protected]
On facets of stable set polytopes of claw-free graphs with stability number three
Electronic Notes in Discrete Mathematics 28 (2007) 185–190 www.elsevier.com/locate/endm
On facets of stable set polytopes of claw-free graphs with stability number three Arnaud Pˆecher 1 Laboratoire Bordelais de Recherche Informatique (LaBRI) 351 cours de la Lib´eration 33405 Talence, France
Pierre Pesneau 2 Math´ematiques Appliqu´ees de Bordeaux (MAB) 351 cours de la Lib´eration 33405 Talence, France
Annegret K. Wagler 3 Otto-von-Guericke-Universit¨at Magdeburg, Fakult¨at f¨ur Mathematik Institut f¨ur Mathematische Optimierung (IMO), Universit¨atsplatz 2, 39106 Magdeburg, Germany
Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a longstanding open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k v∈V1 xv +(k+1) v∈V2 xv ≤ b for some positive integers k and b, and non-empty sets of vertices V1 and V2 . Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.025
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A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients. We prove that the situation is even worse: for every positive integer b, there is a clawfree graph with stability number three whose stable set polytope has a facet with b left coefficients. Keywords: stable set polytope, claw-free graph
The stable set polytope STAB(G) of a graph G is defined as the convex hull of the incidence vectors of all its stable sets. The description of STAB(G) by means of facet-defining inequalities is unknown for most graphs. Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem [5]. A graph is claw-free if the neighborhood of any node does not contain any stable set of size 3. A characterization of the 0/1-valued facets, called rank facets, in stable set polytopes of claw-free graphs was given by Galluccio and Sassano [3]. However, clawfree graphs have non-rank facets in general and even a conjecture regarding their non-rank facets was formulated only recently [8] [9]. Claw-free graphs contain all line graphs, that are graphs obtained by taking the edges of a root graph H as nodes and connecting two nodes of the line graph iff the corresponding edges of H are incident. All facets of the stable set polytope of line graphs are known from matching theory [1], namely, clique constraints and certain rank constraints coming from odd set inequalities. Ben Rebea generalized the odd set inequalities for the matching polytope to clique family inequalities for the stable set polytopes of all graphs. He conjectured (see [7]) and Eisenbrandt et al. [2] recently proved that clique family inequalities suffice for quasi-line graphs, that are graphs where the neighborhood of any node can be partitioned into two cliques. As a consequence, every non-trivial facet of the stable set polytope of quasi-line graphs is of the form k v∈V1 xv + (k + 1) v∈V2 xv ≤ b for some positive integers k and b, and non-empty sets of vertices V1 and V2 . Roughly speaking, this states that the facets of the stable set polytope of quasiline graphs have at most two different left coefficients. The same is true for STAB(G) if α(G) = 2 (Cook 1987, see also [6]) and conjectured if G is claw-free and α(G) ≥ 4 [9]. It is already known that the stable set polytopes of some claw-free graphs G with α(G) = 3 have facets with up to 5 left coefficients [4][6]. 1 2 3
Email:[email protected] Email:[email protected] Email:[email protected]
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
187
In this paper, we prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients. Theorem 0.1 For every positive integer b ≥ 5, there exists a claw free graph Gb with maximum stable set size 3 such that xv + 2 xv + . . . + (b − 1) xv ≤ b (1) v∈V1
v∈V2
v∈Vb−1
is a facet of its stable set polytope such that V1 , . . . , Vb−1 are non-empty sets of vertices. We now proceed to give a description of the graph Gb . The result is based on [8] where two of the authors proved that all non-rank facets associated with clawfree graphs with stability number 3 belong to the following type of inequalities. A co-spanning 1-forest constraint of a graph G with n vertices is a facet of STAB(G) whose n tight stable sets correspond in the complementary graph to the edges of a 1-forest F (consisting of tree- and 1-tree-components, where a 1-tree is a tree with an extra edge) and as many triangles as F has tree-components. Therefore to give a description of Gb , we proceed in 2 steps: •
firstly, we define a graph Hb with clique number 3 and n maximal cliques where n is the number of vertices of Hb ;
•
secondly, we define a graph Hb by adding an appropriate set of edges in order to make the complementary graph Gb of Hb claw-free, with the co-spanning 1forest constraint associated to Hb .
Let k = b − 2 and n = 5k − 3. We define the graph Hb as follows (see Fig. 1): the vertex set V of Hb consists of the n vertices {v0 , v1 , . . . , v4k−6 } ∪ {w0 , w1 , . . . , wk−1 } ∪ {y0 , y1 }, and the graph Hb has exactly n maximal cliques: •
k − 1 triangles U1 , U2 , . . . , Uk−1 where Uj = {wj , v4j−1 , v4j } for every 1 ≤ j ≤ k − 2 and Uk−1 = {w0 , wk−1 , v0 };
•
the edges of a spanning forest made of k − 1 trees T1 , T2 , . . . , Tk−1 where for every 1 ≤ i ≤ k −3, the tree Ti is the chain of size 4 {v4i , v4i+1 , v4i+2 , v4i+3 }, the remaining Tk−2 is the chain of size 7 {v4(k−2) , v4(k−2)+1 , v4k−6 , v0 , v1 , v2 , v3 }, and the remaining tree Tk−1 has vertices {w0 , w1 , . . . , wk−1 } ∪ {y0 , y1 } with edges {y0 w0 , y0 w1 , . . . , y0 wk/2 } ∪ {y1 wk/2 , y1 wk/2+1 , . . . , y1 wk−1 }.
Let c be the function from the vertices of Hb into N defined as follows (see Fig. 1 again):
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A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
c→N i = 0, 2, . . . , 4k − 6 : vi → k − i/4 i = 1, 3, . . . 4k − 7 : vi → i/4 + 2 w0 , w1 , . . . , wk−1 → 1 y0 , y1 → k + 1 Notice that for every edge xy of a tree Ti (1 ≤ i ≤ k−1), we have c(x)+c(y) = k + 2 = b and for every triangle Uj = {x, y, z} (1 ≤ j ≤ k − 1), we have c(x) + c(y) + c(z) = k + 2 = b. It is easy to check that the stable set polytope of the complement of Hb has In. (1) as a facet. v0
v1
v4k−6 v4(k−2)+1
wk−1
v4(k−2) v4(k−2)−1
v2
w0
wk−2
2 v3
w1
v4(k−2)−2 y1
wk−3
y0
w2
v7 v8
v4(k−3)−1
v9
v4(k−3)−2
1
v4k/2
k−1 1
3 k−1
1
3
k+1
1
k+1
3 k−2
k−2 4
4 1
wk/2
v4k/2+1
2
k−1
v6
v4(k−3)
k
1
3
v5
v4(k−3)+1
2
1
k−1
v4
k
2
k
v4k/2−2 v4k/2−1
graph Hb
k−i+1
i+2 k−i
i+1
coefficients c(v)
Fig. 1. The spanning 1-forest constraint Hb and the coefficients of the associated facet
We now proceed to the second step: we need to add edges to Hb in order to make the complementary graph claw-free without violating In. (1) or increasing the clique number. Let Hb be the graph with vertex set V and edge set E ⊃ E defined as follows (Fig. 2 depicts the additional edges for i = 2 below and b = 9): E = E
({v4i v1 , v4i v3 , . . . , v4i v4i−3 } ∪ {v4i wi+1 , v4i wi+2 , . . . , v4i wk−1 })
i=1,...,k−2
({v4i−1 v4i+2 , v4i−1 v4i+4 , . . . , v4i−1 v4k−6 } ∪ {v4i−1 w0 , v4i−1 w1 , . . . , v4i−1 wi−1 })
i=1,...,k−2
({wi v0 , wi v2 , . . . , wi v4i−4 } ∪ {wi v4i+3 , wi v4i+5 , . . . , wi v4k−7 } ∪ {wi y0 , wi y1 })
i=1,...,k−2
Let Gb be the complement of Hb and define
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
189
V1 = {w0 , w1 , . . . , wk−1 } V2 = {v1 , v3 , v4k−8 , v4k−6 } V3 = {v5 , v7 , v4k−12 , v4k−10 } .. .. .. . . . Vi = {v4i−7 , v4i−5 , v4k−4i , v4k−4i+2 } .. .. .. . . . Vk−1 = {v4k−11 , v4k−9 , v4 , v6 } Vk = {v0 , v2 , v4k−7 } Vk+1 = {y0 , y1 } Notice that V1 , . . . , Vk+1 is a partition of the vertex set of Gb into non-empty sets such that for every 1 ≤ j ≤ k + 1 and every v ∈ Vj , c(v) = j. Theorem 0.1 asserts that xv + 2 xv + . . . + (b − 1) xv ≤ b v∈V1
v∈V2
v∈Vb−1
is indeed a facet of the stable set polytope of the claw-free graph Gb with stability number 3. 7
2
7
2 7
2
1
1
2
6
6 1
1
3 6
3
8
8
6
1
1
3
3
1 5
5 4
4 5
4
4
5
Fig. 2. The spanning 1-forest constraint H9 , the coefficients of the associated facet and some of the additional edges
References [1] Edmonds, J., Maximum matching and a polyhedron with (0,1) vertices, Journal Res. Nat. Bur. Standards 69B (1965), pp. 125–130.
190
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
[2] Eisenbrandt, F., G. Oriolo, G. Stauffer and P. Ventura, Circular one matrices and the stable set polytope of quasi-line graphs, Lecture Notes in Computer Science 3509 (2005), pp. 291–305. [3] Galluccio, A. and A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Comb. Theory B 69 (1997), pp. 1–38. [4] Giles, R. and j. L.E. Trotter, On stable set polyhedra for K1,3 -free graphs, J. Comb. Theory B 31 (1981), pp. 313–326. [5] Gr¨otschel, M., L. Lov´asz and A. Schrijver, “Geometric Algorithms and Combinatorial Optimization,” Springer-Verlag, 1988. [6] Liebling, T., G. Oriolo, B. Spille and G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Mathematical Methods of Operations Research 59 (2004), pp. 25–35. [7] Oriolo, G., On the stable set polytope for quasi-line graphs, In: Special issue on stability problems, Discrete Applied Math. 132 (2003), pp. 185–201. [8] Pˆecher, A. and A. Wagler, A conjecture on the stable set polytope for claw-free graphs, preprint 11-2006, Otto-von-Guericke-Universit¨at Magdeburg, Fakult¨at f¨ur Mathematik. [9] Stauffer, G., “On the Stable Set Polytope of Claw-free graphs,” Ph.D. thesis, EPF Lausanne (2005).
On facets of stable set polytopes of claw-free graphs with stability number three Arnaud Pˆecher 1 Laboratoire Bordelais de Recherche Informatique (LaBRI) 351 cours de la Lib´eration 33405 Talence, France
Pierre Pesneau 2 Math´ematiques Appliqu´ees de Bordeaux (MAB) 351 cours de la Lib´eration 33405 Talence, France
Annegret K. Wagler 3 Otto-von-Guericke-Universit¨at Magdeburg, Fakult¨at f¨ur Mathematik Institut f¨ur Mathematische Optimierung (IMO), Universit¨atsplatz 2, 39106 Magdeburg, Germany
Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a longstanding open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k v∈V1 xv +(k+1) v∈V2 xv ≤ b for some positive integers k and b, and non-empty sets of vertices V1 and V2 . Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.025
186
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients. We prove that the situation is even worse: for every positive integer b, there is a clawfree graph with stability number three whose stable set polytope has a facet with b left coefficients. Keywords: stable set polytope, claw-free graph
The stable set polytope STAB(G) of a graph G is defined as the convex hull of the incidence vectors of all its stable sets. The description of STAB(G) by means of facet-defining inequalities is unknown for most graphs. Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem [5]. A graph is claw-free if the neighborhood of any node does not contain any stable set of size 3. A characterization of the 0/1-valued facets, called rank facets, in stable set polytopes of claw-free graphs was given by Galluccio and Sassano [3]. However, clawfree graphs have non-rank facets in general and even a conjecture regarding their non-rank facets was formulated only recently [8] [9]. Claw-free graphs contain all line graphs, that are graphs obtained by taking the edges of a root graph H as nodes and connecting two nodes of the line graph iff the corresponding edges of H are incident. All facets of the stable set polytope of line graphs are known from matching theory [1], namely, clique constraints and certain rank constraints coming from odd set inequalities. Ben Rebea generalized the odd set inequalities for the matching polytope to clique family inequalities for the stable set polytopes of all graphs. He conjectured (see [7]) and Eisenbrandt et al. [2] recently proved that clique family inequalities suffice for quasi-line graphs, that are graphs where the neighborhood of any node can be partitioned into two cliques. As a consequence, every non-trivial facet of the stable set polytope of quasi-line graphs is of the form k v∈V1 xv + (k + 1) v∈V2 xv ≤ b for some positive integers k and b, and non-empty sets of vertices V1 and V2 . Roughly speaking, this states that the facets of the stable set polytope of quasiline graphs have at most two different left coefficients. The same is true for STAB(G) if α(G) = 2 (Cook 1987, see also [6]) and conjectured if G is claw-free and α(G) ≥ 4 [9]. It is already known that the stable set polytopes of some claw-free graphs G with α(G) = 3 have facets with up to 5 left coefficients [4][6]. 1 2 3
Email:[email protected] Email:[email protected] Email:[email protected]
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
187
In this paper, we prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients. Theorem 0.1 For every positive integer b ≥ 5, there exists a claw free graph Gb with maximum stable set size 3 such that xv + 2 xv + . . . + (b − 1) xv ≤ b (1) v∈V1
v∈V2
v∈Vb−1
is a facet of its stable set polytope such that V1 , . . . , Vb−1 are non-empty sets of vertices. We now proceed to give a description of the graph Gb . The result is based on [8] where two of the authors proved that all non-rank facets associated with clawfree graphs with stability number 3 belong to the following type of inequalities. A co-spanning 1-forest constraint of a graph G with n vertices is a facet of STAB(G) whose n tight stable sets correspond in the complementary graph to the edges of a 1-forest F (consisting of tree- and 1-tree-components, where a 1-tree is a tree with an extra edge) and as many triangles as F has tree-components. Therefore to give a description of Gb , we proceed in 2 steps: •
firstly, we define a graph Hb with clique number 3 and n maximal cliques where n is the number of vertices of Hb ;
•
secondly, we define a graph Hb by adding an appropriate set of edges in order to make the complementary graph Gb of Hb claw-free, with the co-spanning 1forest constraint associated to Hb .
Let k = b − 2 and n = 5k − 3. We define the graph Hb as follows (see Fig. 1): the vertex set V of Hb consists of the n vertices {v0 , v1 , . . . , v4k−6 } ∪ {w0 , w1 , . . . , wk−1 } ∪ {y0 , y1 }, and the graph Hb has exactly n maximal cliques: •
k − 1 triangles U1 , U2 , . . . , Uk−1 where Uj = {wj , v4j−1 , v4j } for every 1 ≤ j ≤ k − 2 and Uk−1 = {w0 , wk−1 , v0 };
•
the edges of a spanning forest made of k − 1 trees T1 , T2 , . . . , Tk−1 where for every 1 ≤ i ≤ k −3, the tree Ti is the chain of size 4 {v4i , v4i+1 , v4i+2 , v4i+3 }, the remaining Tk−2 is the chain of size 7 {v4(k−2) , v4(k−2)+1 , v4k−6 , v0 , v1 , v2 , v3 }, and the remaining tree Tk−1 has vertices {w0 , w1 , . . . , wk−1 } ∪ {y0 , y1 } with edges {y0 w0 , y0 w1 , . . . , y0 wk/2 } ∪ {y1 wk/2 , y1 wk/2+1 , . . . , y1 wk−1 }.
Let c be the function from the vertices of Hb into N defined as follows (see Fig. 1 again):
188
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
c→N i = 0, 2, . . . , 4k − 6 : vi → k − i/4 i = 1, 3, . . . 4k − 7 : vi → i/4 + 2 w0 , w1 , . . . , wk−1 → 1 y0 , y1 → k + 1 Notice that for every edge xy of a tree Ti (1 ≤ i ≤ k−1), we have c(x)+c(y) = k + 2 = b and for every triangle Uj = {x, y, z} (1 ≤ j ≤ k − 1), we have c(x) + c(y) + c(z) = k + 2 = b. It is easy to check that the stable set polytope of the complement of Hb has In. (1) as a facet. v0
v1
v4k−6 v4(k−2)+1
wk−1
v4(k−2) v4(k−2)−1
v2
w0
wk−2
2 v3
w1
v4(k−2)−2 y1
wk−3
y0
w2
v7 v8
v4(k−3)−1
v9
v4(k−3)−2
1
v4k/2
k−1 1
3 k−1
1
3
k+1
1
k+1
3 k−2
k−2 4
4 1
wk/2
v4k/2+1
2
k−1
v6
v4(k−3)
k
1
3
v5
v4(k−3)+1
2
1
k−1
v4
k
2
k
v4k/2−2 v4k/2−1
graph Hb
k−i+1
i+2 k−i
i+1
coefficients c(v)
Fig. 1. The spanning 1-forest constraint Hb and the coefficients of the associated facet
We now proceed to the second step: we need to add edges to Hb in order to make the complementary graph claw-free without violating In. (1) or increasing the clique number. Let Hb be the graph with vertex set V and edge set E ⊃ E defined as follows (Fig. 2 depicts the additional edges for i = 2 below and b = 9): E = E
({v4i v1 , v4i v3 , . . . , v4i v4i−3 } ∪ {v4i wi+1 , v4i wi+2 , . . . , v4i wk−1 })
i=1,...,k−2
({v4i−1 v4i+2 , v4i−1 v4i+4 , . . . , v4i−1 v4k−6 } ∪ {v4i−1 w0 , v4i−1 w1 , . . . , v4i−1 wi−1 })
i=1,...,k−2
({wi v0 , wi v2 , . . . , wi v4i−4 } ∪ {wi v4i+3 , wi v4i+5 , . . . , wi v4k−7 } ∪ {wi y0 , wi y1 })
i=1,...,k−2
Let Gb be the complement of Hb and define
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
189
V1 = {w0 , w1 , . . . , wk−1 } V2 = {v1 , v3 , v4k−8 , v4k−6 } V3 = {v5 , v7 , v4k−12 , v4k−10 } .. .. .. . . . Vi = {v4i−7 , v4i−5 , v4k−4i , v4k−4i+2 } .. .. .. . . . Vk−1 = {v4k−11 , v4k−9 , v4 , v6 } Vk = {v0 , v2 , v4k−7 } Vk+1 = {y0 , y1 } Notice that V1 , . . . , Vk+1 is a partition of the vertex set of Gb into non-empty sets such that for every 1 ≤ j ≤ k + 1 and every v ∈ Vj , c(v) = j. Theorem 0.1 asserts that xv + 2 xv + . . . + (b − 1) xv ≤ b v∈V1
v∈V2
v∈Vb−1
is indeed a facet of the stable set polytope of the claw-free graph Gb with stability number 3. 7
2
7
2 7
2
1
1
2
6
6 1
1
3 6
3
8
8
6
1
1
3
3
1 5
5 4
4 5
4
4
5
Fig. 2. The spanning 1-forest constraint H9 , the coefficients of the associated facet and some of the additional edges
References [1] Edmonds, J., Maximum matching and a polyhedron with (0,1) vertices, Journal Res. Nat. Bur. Standards 69B (1965), pp. 125–130.
190
A. Pêcher et al. / Electronic Notes in Discrete Mathematics 28 (2007) 185–190
[2] Eisenbrandt, F., G. Oriolo, G. Stauffer and P. Ventura, Circular one matrices and the stable set polytope of quasi-line graphs, Lecture Notes in Computer Science 3509 (2005), pp. 291–305. [3] Galluccio, A. and A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Comb. Theory B 69 (1997), pp. 1–38. [4] Giles, R. and j. L.E. Trotter, On stable set polyhedra for K1,3 -free graphs, J. Comb. Theory B 31 (1981), pp. 313–326. [5] Gr¨otschel, M., L. Lov´asz and A. Schrijver, “Geometric Algorithms and Combinatorial Optimization,” Springer-Verlag, 1988. [6] Liebling, T., G. Oriolo, B. Spille and G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Mathematical Methods of Operations Research 59 (2004), pp. 25–35. [7] Oriolo, G., On the stable set polytope for quasi-line graphs, In: Special issue on stability problems, Discrete Applied Math. 132 (2003), pp. 185–201. [8] Pˆecher, A. and A. Wagler, A conjecture on the stable set polytope for claw-free graphs, preprint 11-2006, Otto-von-Guericke-Universit¨at Magdeburg, Fakult¨at f¨ur Mathematik. [9] Stauffer, G., “On the Stable Set Polytope of Claw-free graphs,” Ph.D. thesis, EPF Lausanne (2005).