Publication

Extension complexity of stable set polytopes of bipartite graphs

Yuri Faenza, Manuel Francesco Aprile
2017
Conference Papers

Abstract

The extension complexity xc(P) of a polytope P is the minimum number of facets of a polytope that affinely projects to P. Let G be a bipartite graph with n vertices, m edges, and no isolated vertices. Let STAB(G) be the convex hull of the stable sets of G. It is easy to see that n⩽xc(STAB(G))⩽n+m. We improve both of these bounds. For the upper bound, we show that xc(STAB(G)) is O(n2logn), which is an improvement when G has quadratically many edges. For the lower bound, we prove that xc(STAB(G)) is Ω(nlogn) when G is the incidence graph of a finite projective plane. We also provide examples of 3-regular bipartite graphs G such that the edge vs stable set matrix of G has a fooling set of size |E(G)|.

Official source

https://infoscience.epfl.ch/entities/publication/6812a473-a707-4121-964d-c4766eafe4f8

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Yuri Faenza, Manuel Francesco Aprile
2017
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/6812a473-a707-4121-964d-c4766eafe4f8

About this result

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (31)

Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in red, each edge has endpoints of differing colors, as is required in the graph coloring problem.

Complete bipartite graph

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher.

Biregular graph

In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in is and the degree of the vertices in is , then the graph is said to be -biregular. Every complete bipartite graph is -biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular. An -biregular graph must satisfy the equation .

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