6,917 research outputs found
The extremal genus embedding of graphs
Let Wn be a wheel graph with n spokes. How does the genus change if adding a
degree-3 vertex v, which is not in V (Wn), to the graph Wn? In this paper,
through the joint-tree model we obtain that the genus of Wn+v equals 0 if the
three neighbors of v are in the same face boundary of P(Wn); otherwise,
{\deg}(Wn + v) = 1, where P(Wn) is the unique planar embedding of Wn. In
addition, via the independent set, we provide a lower bound on the maximum
genus of graphs, which may be better than both the result of D. Li & Y. Liu and
the result of Z. Ouyang etc: in Europ. J. Combinatorics. Furthermore, we obtain
a relation between the independence number and the maximum genus of graphs, and
provide an algorithm to obtain the lower bound on the number of the distinct
maximum genus embedding of the complete graph Km, which, in some sense,
improves the result of Y. Caro and S. Stahl respectively
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Vertex Splitting and Upper Embeddable Graphs
The weak minor G of a graph G is the graph obtained from G by a sequence of
edge-contraction operations on G. A weak-minor-closed family of upper
embeddable graphs is a set G of upper embeddable graphs that for each graph G
in G, every weak minor of G is also in G. Up to now, there are few results
providing the necessary and sufficient conditions for characterizing upper
embeddability of graphs. In this paper, we studied the relation between the
vertex splitting operation and the upper embeddability of graphs; provided not
only a necessary and sufficient condition for characterizing upper
embeddability of graphs, but also a way to construct weak-minor-closed family
of upper embeddable graphs from the bouquet of circles; extended a result in J:
Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of
determining the upper embeddability of a graph can be reduced much by the
results obtained in this paper
Genus Ranges of 4-Regular Rigid Vertex Graphs
We introduce a notion of genus range as a set of values of genera over all
surfaces into which a graph is embedded cellularly, and we study the genus
ranges of a special family of four-regular graphs with rigid vertices that has
been used in modeling homologous DNA recombination. We show that the genus
ranges are sets of consecutive integers. For any positive integer , there
are graphs with vertices that have genus range for all
, and there are graphs with vertices with genus range
for all or . Further, we show that
for every there is such that is a genus range for graphs with
and vertices for all . It is also shown that for every ,
there is a graph with vertices with genus range , but there
is no such a graph with vertices
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