Closed 2-cell embeddings of graphs with no V8-minors (original) (raw)

Closed 2-cell embeddings of graphs with no -minors

Discrete Mathematics, 2001

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.

Closed 2-cell embeddings of 4 cross-cap embeddable graphs

Discrete Mathematics, 1996

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a circuit in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. A graph is called k cross-cap embeddable if it can be embedded in the non-orientable surface of k cross-caps. In this paper, we prove that every 2-connected 4 cross-cap embeddable graph G has a closed 2-cell embedding in some surface. As a corollary, G has a cycle double cover, i.e., G has a set of circuits containing every edge exactly twice.

Orientable embeddings and orientable cycle double covers

2009

In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2-edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.

A common cover of graphs and 2-cell embeddings

Journal of Combinatorial Theory, Series B, 1986

Let G and H be finite graphs with equal uniform degree refinements. Their finite common covering graph G 0 H is constructed. It is shown that G, H, and G 0 H can be 2-cell embedded in orientable surfaces M, N and S", respectively, in such a way that the graph covering projections G 0 H + G and G 0 H + H extend to branched coverings M + S-+ N of the surfaces. Additional properties of G 0 H are used to obtain some nontrivial consequences about coverings of some planar graphs.

Orientable embeddings and orientable cycle double covers of projective-planar graphs

European Journal of Combinatorics, 2011

Closed 2-cell Embeddings of 5-crosscap Embeddable Graphs

European Journal of Combinatorics, 1997

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i. e. an embedding that each face is bounded by a circuit in the graph. A graph is called k-crosscap embeddable if it can be embedded in the surface of non-orientable genus k. We confirm the strong embedding conjecture for 5-crosscap embed-461

Obstructions for two-vertex alternating embeddings of graphs in surfaces

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

2-Cell Embeddings with Prescribed Face Lengths and Genus

Annals of Combinatorics, 2010

Let n be a positive integer, let d 1 ,. .. , d n be a sequence of positive integers, and let q = 1 2 n i=1 d i. It is shown that there exists a connected graph G of order n, whose degree sequence is d 1 ,. .. , d n and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n − q + 1, if and only if q is an integer and q ≥ n − 1. Moreover, the graph G is loopless if and only if d i ≤ q for i = 1,. .. , n. This, in particular, answers a question of Arkadiy Skopenkov.

A characterization of K2,4K_{2,4}K2,4-minor-free graphs

We provide a complete structural characterization of K2,4K_{2,4}K2,4-minor-free graphs. The 333-connected K2,4K_{2,4}K2,4-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K4K_4K4 and, for each nge5n \ge 5nge5, 2n−82n-82n−8 nonisomorphic graphs of order nnn. To describe the 222-connected K2,4K_{2,4}K2,4-minor-free graphs we use xyxyxy-outerplanar graphs, graphs embeddable in the plane with a Hamilton xyxyxy-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, xyxyxy-outerplanar graphs are precisely the graphs that have no rooted K2,2K_{2,2}K2,2-minor where xxx and yyy correspond to the two vertices on one side of the bipartition of K2,2K_{2,2}K2,2. Each 222-connected K2,4K_{2,4}K2,4-minor-free graph is then (i) outerplanar, (ii) the union of three xyxyxy-outerplanar graphs and possibly the edge xyxyxy, or (iii) obtained from a 333-connected K2,4K_{2,4}K2,4-minor-free graph by replacing each edge $x_iy_...

Closed 2-cell embeddings of graphs with no V8-minors (original) (raw)

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