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A Characterization of Seymour Graphs A. A. Ageev A. V. Kostochka Z. Szigeti y
 

Summary: A Characterization of Seymour Graphs
A. A. Ageev  A. V. Kostochka  Z. Szigeti y
Abstract
Following Gerards [1] we say that a connected undirected graph G is a Seymour
graph if the maximum number of edge disjoint T-cuts is equal to the cardinality
of a minimum T-join for every even subset T of V (G). Several families of graphs
have been shown to be subfamilies of Seymour graphs (Seymour [4][5], Gerards [1],
Szigeti [6]). In this paper we prove a characterization of Seymour graphs which was
conjectured by Seb}o and implies the results mentioned above.
1 Introduction
T. Graphs in this paper are undirected, connected, and may have loops and multiple
edges.
Let G be a graph. For F  E(G) and x; y 2 V (G), we write xy 2 F if some edge of
G with endpoints x and y is in F .
For X  V (G), the cut (X) is the set of edges connecting X and V (G)nX, N(X) =
fv 2 V (G) n X : v has neighbors in Xg. If X = fxg we write (x), N(x). For F  E(G)
and v 2 V (G), let d F (v) denote the degree of v in the subgraph of G spanned by F . A
pair (G; T ) where T is an even subset of V (G) is called a graft.
Let T be an even subset of V (G). If jX \ T j is odd the cut (X) is called a T-cut.
A set F of edges of G is a T-join if T = fv 2 V (G) j d F (v) is oddg. Let (G; T ) denote

  

Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

 

Collections: Mathematics