William R. Pulleyblank

The perfectly matchable subgraph polytope of a bipartite graph

Abstract : The following type of problem arises in practice: in a node-weighted graph G, find a minimum weight node set that satisfies certain conditions and, in addition, induces a perfectly matchable subgraph of G. This has led us to study the convex hull of incidence vectors of node sets that induce perfectly matchable subgraphs of a graph G, which we call the perfectly matchable subgraph polytype of G. For the case when G is bipartite, we give a linear characterization of this polytype, i.e., specify a system of linear inequalities whose basic solutions are the incidence vectors of perfectly matchable node sets of G. We derive this result by three different approaches, using linear programming duality, projection, and lattice polyhedra, respectively. The projection approach is used here for the first time as a proof method in polyhedral combinatorics, and seems to have many similar applications. Finally, we completely characterize the facets of our polytype, i.e., separate the essential inequalities of our linear defining system from the redundant ones. (Author)

A matching problem with side conditions

Given a graph G, a 2-matching is an assignment of nonnegative integers to the edges of G such that for each node i of G, the sum of the values on the edges incident with i is at most 2. A triangle-free 2-matching is a 2-matching such that no cycle of size 3 in G has the value 1 assigned to all of its edges. In this paper we describe explicity the convex hull of triangle-free 2-matchings by means of its extreme points and of its facets. We give a polynomially bounded algorithm which maximizes a linear function over the set of triangle-free 2-matchings. Finally we discuss some related problems.

A note on graphs spanned by Eulerian graphs

We show that the problem raised by Boesch, Suffel, and Tindell of determining whether or not a graph is spanned by an Eulerian subgraph is NP-complete. We also note that there does exist a good algorithm for determining if a graph is spanned by a subgraph having positive even degree at every node.

On partitioning the edges of graphs into connected subgraphs

For any positive integer s, an s-partition of a graph G = (V, E) is a partition of E into E1 ∪ E2 ∪… ∪ Ek, where ∣Ei∣ = s for 1 ≤ i ≤ k − 1 and 1 ≤ ∣Ek∣ ≤ s and each Ei induces a connected subgraph of G. We prove n n(i) nIf G is connected, then there exists a 2-partition, but not necessarily a 3-partition; n n(ii) nIf G is 2-edge connected, then there exists a 3-partition, but not necessarily a 4-partition; n n(iii) nIf G is 3-edge connected, then there exists a 4-partition; n n(iv) nIf G is 4-edge connected, then there exists an s-partition for all s.

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.

Hamiltonicity in (0–1)-polyhedra☆

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ⊆ Rn is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ⊆ Rn has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ⊆ Rd having (0–1)-valued vertices such that G(P) ⋍ G(P′). Some combinatorial consequences of these results are also discussed.

Minimum node covers and 2-bicritical graphs

The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical and hence the linear relaxation almost never helps for large random graphs.

The traveling salesman problem in graphs with 3-edge cutsets

This paper analyzes decomposition properties of a graph that, when they occur, permit a polynomial solution of the traveling salesman problem and a description of the traveling salesman polytope by a system of linear equalities and inequalities. The central notion is that of a 3-edge cutset, namely, a set of 3 edges that, when removed, disconnects the graph. Conversely, our approach can be used to construct classes of graphs for which there exists a polynomial algorithm for the traveling salesman problem. The approach is illustrated on two examples, Halin graphs and prismatic graphs.

Hamiltonicity and combinatorial polyhedra

Abstract We say that a polyhedron with 0–1 valued vertices is combinatorial if the midpoint of the line joining any pair of nonadjacent vertices is the midpoint of the line joining another pair of vertices. We show that the class of combinatorial polyhedra includes such well-known classes of polyhedra as matching polyhedra, matroid basis polyhedra, node packing or stable set polyhedra and permutation polyhedra. We show the graph of a combinatorial polyhedron is always either a hypercube (i.e., isomorphic to the convex hull of a k-dimension unit cube) or else is hamilton connected (every pair of nodes is the set of terminal nodes of a hamilton path). This implies several earlier results concerning special cases of combinatorial polyhedra.

Matchings in regular graphs

We consider k-regular graphs with specified edge connectivity and show how some classical theorems and some new results concerning the existence of matchings in such graphs can be proved by using the polyhedral characterization of Edmonds. In addition, we show that lower bounds of Lovasz and Plummer on the number of perfect matchings in bicritical graphs can be improved for cubic bicritical graphs.