Donald K. Wagner

The dominant of the 2-connected-Steiner-subgraph polytope for W 4 -free graphs

Abstract This paper presents a linear-inequality description of the dominant of the polytope of the 2-connected Steiner subgraphs of a given W4-free graph. For the special case of 2-connected spanning subgraphs, a description of the polytope is given. The latter contains the Traveling-Salesman polytope for W4-free graphs as a face.

The arborescence-realization problem

A {0, 1}-matrix M is arborescence graphic if there exists an arborescence T such that the arcs of T are indexed on the rows of M and the columns of M are the incidence vectors of the arc sets of dipaths of T. If such a T exists, then T is an arborescence realization for M. This paper presents an almost-linear-time algorithm to determine whether a given {0, 1}-matrix is arborescence graphic and, if so, to construct an arborescence realization. The algorithm is then applied to recognize a subclass of the extended-Horn satisfiability problems introduced by Chandru and Hooker (1991).

Linear-time Algorithms for the 2-connected Steiner Subgraph Problem On Special Classes of Graphs

The 2-connected Steiner subgraph problem is that of finding a minimum-weight 2-connected subgraph that spans a subset of distinguished vertices. This paper presents linear-time algorithms for solving the 2-connected Steiner subgraph problem on two special classes of graphs, W4-free graphs and Halin graphs. Although different in detail, the algorithms adopt a common strategy exploiting known decompositions. As a special case, the algorithms also solve the Traveling Salesman Problem on W4-free graphs and Halin graphs.

A polynomial-time simplex method for the maximum k -flow problem

A generalization of the maximum-flow problem is considered in which every unit of flow sent from the source to the sink yields a payoff of

Decomposition of 3-connected graphs

k. In addition, the capacity of any arce can be increased at a per-unit cost of

Delta-wye reduction of almost-planar graphs

ce. The problem is to determine how much arc capacity to purchase for each arc and how much flow to send so as to maximize the net profit. This problem can be modeled as a circulation problem. The main result of this paper is that this circulation problem can be solved by the network simplex method in at mostkmn pivots. Whence = 1 for each arce, this yields a strongly polynomial-time simplex method. This result uses and extends a result of Goldfarb and Hao which states that the standard maximum-flow problem can be solved by the network simplex method in at mostmn pivots.

On Mighton's characterization of graphic matroids

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K3,n for somen≥3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.

Weakly 3-Connected Graphs

A non-planar graph G is almost planar if, for every edge e of G , either G ? e or G / e is planar. The main result of this paper is that every almost-planar graph is delta-wye reducible to K 3 , 3 , and moreover, there exists a reduction sequence in which every graph is almost planar. Analogous results are shown to hold for other classes of graphs, and also for regular, almost-graphic matroids.

Uncovering generalized-network structure in matrices

Mighton (2008) [5] recently gave a new characterization of graphic matroids. This note combines Mightons approach with a result of Cunningham (1982) [4] to provide a shorter, more direct proof of Mightons result.

Vertex 2-isomorphism

A 3-connected graph