4,762 research outputs found
Removable Edges in 4-Connected Graphs
Research on structural characterizations of graphs is a very popular topic in graph theory. The concepts of contractible edges and removable edges of graphs are powerful tools to study the structure of graphs and to prove properties of graphs by induction. \ud
In 1998, Yin gave a convenient method to construct 4-connected graphs by using the existence of removable edges and contractible edges. He showed that a 4-connected graph can be obtained from a 2-cyclic graph by the following four operations: (i) adding edges, (ii) splitting vertices, (iii) adding vertices and removing edges, and (iv) extending vertices. Based on the above operations, we gave the following definition of removable edges in 4-connected graphs. Definition: Let be a 4-connected graph. For an edge of , we perform the following operations on : First, delete the edge from , resulting in the graph ; Second, for each vertex of degree 3 in , delete from and then completely connect the 3 neighbors of by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by . If is still 4-connected, then the edge is called "removable"; otherwise, is called "unremovable". In the thesis we get the following results: (1) we study how many removable edges may exist in a cycle of a 4-connected graph, and we give examples to show that our results are in some sense the best possible. (2) We obtain results on removable edges in a longest cycle of a 4-connected graph. We also show that for a 4-connected graph of minimum degree at least 5 or girth at least 4, any edge of is removable or contractible. (3) We study the distribution of removable edges on a Hamilton cycle of a 4-connected graph, and show that our results cannot be improved in some sense. (4) We prove that every 4-connected graph of order at least six except 2-cyclic graph with order 6 has at least removable edges. We also give a structural characterization of 4-connected graphs for which the lower bound is sharp. (5) We study how many removable edges there are in a spanning tree of a 4-connected graph and how many removable edges exist outside a cycle of a 4-connected graph. We also give examples to show that our results can not be improved in some sense
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible, non-separating and noncontractible separating Hamiltonian cycle
in the edge graph of polyhedral maps on surfaces. In particular, we show the
existence of contractible Hamiltonian cycle in equivelar triangulated maps. We
also present an algorithm to construct such cycles whenever it exists.Comment: 14 page
- …