On the convex hull of 3-cycles of the complete graph

Abstract

Let n K be the complete undirected graph with n vertices. A 3-cycle is a cycle consisting of 3 edges.The 3-cycle polytope is defined as the convex hull of the incidence vectors of all 3-cycles in n K . In this paper, we present a polyhedral analysis of the 3-cycle polytope. In particular, we give several classes of facet defining inequalities of this polytope and we prove that the separation problem associated to one of these classes of inequalities is NP-complete. Finally, it is proved that the 3-cycle polytope is a 2-neighborly polytope.

On cardinality constrained cycle and path polytopes

“…is valid for P When n ≥ 6, the claim follows from Proposition 2 of Kovalev, Maurras, and Vaxés [16], Proposition 2 of Maurras and Nguyen [18], and the fact that m ≥ 2.…”

“…It was shown in [16] and [18] that dim P (p) C (K n ) = |E| − 1 for 3 ≤ p ≤ n − 1 and n ≥ 5. Thus, it is easy to verify that dim P c C (K n ) = |E| = n(n − 1)/2 for all n ≥ 4, since m ≥ 2.…”

“…In this section, we consider the undirected cardinality constrained cycle polytope P c C (K n ) defined on the complete graph K n = (N, E), where c is a cardinality sequence with 3 ≤ c 1 < • • • < c m ≤ n and m ≥ 2. It was shown in [16] and [18] that dim P Facet defining inequalities for P c C (K n ) can be derived directly from the inequalities mentioned in Corollary 4.1 (b)-(h), since these inequalities are equivalent to symmetric inequalities. A valid inequality cx ≤ γ for P c C (D n ) is said to be symmetric if c ij = c ji holds for all i < j.…”

A Facet Defining of the Dicycle Polytope

“…In the case of undirected graphs, the cycle polytope has been studied by Coullard and Pulleyblank [6], and after by Bauer [3]. Kovalev et al [15] and Bauer et al [4] study the cardinality constrained cycle polytope which is the convex hull of all cycles with at most p nodes on a complete undirected graph. The p-cycle polytope has been also studied by Nguyen and Maurras [17,18] for p = 3 and for 2 < p < n.…”