The imbedding index of a graph (original) (raw)
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Flexibility of Polyhedral Embeddings of Graphs in Surfaces
Journal of Combinatorial Theory, Series B, 2001
Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there is a function ξ : N 0 → N 0 such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ≥ 3 into surfaces whose Euler genus is at most g.
Combinatorial Local Planarity and the Width of Graph Embeddings
Canadian Journal of Mathematics, 1992
Let G be a graph embedded in a closed surface. The embedding is “locally planar” if for each face, a “large” neighbourhood of this face is simply connected. This notion is formalized, following [RV], by introducing the width ρ(ψ) of the embedding ψ. It is shown that embeddings with ρ(ψ) ≥ 3 behave very much like the embeddings of planar graphs in the 2-sphere. Another notion, “combinatorial local planarity”, is introduced. The criterion is independent of embeddings of the graph, but it guarantees that a given cycle in a graph G must be contractible in any minimal genus embedding of G (either orientable, or non-orientable). It generalizes the width introduced before. As application, short proofs of some important recently discovered results about embeddings of graphs are given and generalized or improved. Uniqueness and switching equivalence of graphs embedded in a fixed surface are also considered.
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Acta Mathematica Scientia, 2002
In this paper, the authors discuss the upper bound for the genus of strong embeddings for 3-connected planar graphs on higher surfaces. It is shown that the problem of determining the upper bound for the strong embedding of 3-connected planar neartriangulations on higher non-orientable surfaces is NP-hard. As a corollary, a theorem of Richter, Seymour and Siran about the strong embedding of 3-connected planar graphs is generalized to orientable surface.
A Subset S of the vertex set of a graph G is called a dominating set of G, if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = D 1 D 2 D k of the vertex set of G is said to be a domatic partition or simply a d-partition of G, if each class of D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d G . A graph G is said to be domatically critical or d-critical if for every edge x in G, d G − x < d G otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of d-critical graph H containing G as an induced sub graph denoted by G . In this paper, we find the upper bound of G for few well known classes of graphs.
Embedding index in some classes of graphs
A Subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = {D 1 , D 2 , …, D k } of the vertex set of G is said to be a domatic partition or simply a d-partition of G if each class D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d (G). A graph G is said to be domatically critical or d-critical if for every edge x in G, d (G -x) < d (G), otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of a d-critical graph H containing G as an induced subgraph denoted by ) (G θ . In this paper, we find the ) (G θ for the Barbell graph, the Lollipop graph and the Tadpole graph.
An obstruction to embedding graphs in surfaces
Discrete Mathematics, 1989
It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.
In-trees and plane embeddings of outerplanar graphs
Bit Numerical Mathematics, 1990
A notion of an in-tree is introduced. It is then used to characterize and count plane embeddings of outerplanar graphs. In-trees have also been applied in the study of independent vertex covers of faces in outerplanar graphs.
The enumeration of planar graphs via Wick's theorem
Advances in Mathematics, 2009
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type partition function. Some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover. We express the number of the graphs with a fixed directed cycle double cover as the Gaussian matrix integral of an Ihara-Selberg-type function.