Rooted K4-Minors

List-coloring graphs without K4, k-minors

In this note, it is shown that every graph with no K4,k-minor is 4k-list-colorable. We also give an extremal function for the existence for a K4,k-minor. Our proof implies that there is a linear time algorithm for showing that either G has a K4,k-minor or G is 4k-choosable. In fact, if the latter holds, then the algorithm gives rise to a 4k-list-coloring. © 2008 Elsevier B.V. All rights reserved.

Excluded-minor Characterization of Apex-outerplanar

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: K4 and K2,3. The class of graphs that contain a vertex whose removal leaves an outerplanar graph is also minor-closed. We provide the complete list of 57 excluded minors for this class.

The degree distribution in unlabelled 2-connected graph families

An outerplanar graph is a planar graph which can be embedded in the plane such that all nodes lie on the outer face. A series-parallel graph is a graph which is the result of series (subdivision) or parallel (doubling) extensions of the edges of a forest. (Both graph families can be described via families of forbidden minors {K4,K23} and {K4}, respectively, where K4 denotes the complete graph o...

B . A . Reed Graph Minors I : Rooted Routing

Normal Binary Graph Models

We show that the marginal semigroup of a binary graph model is normal if and only if the graph is free of K4 minors. The technique, based on the interplay of normality and the geometry of the marginal cone, has potential applications to other normality questions in algebraic statistics.