837 research outputs found
Counting unlabelled toroidal graphs with no K33-subdivisions
We provide a description of unlabelled enumeration techniques, with complete
proofs, for graphs that can be canonically obtained by substituting 2-pole
networks for the edges of core graphs. Using structure theorems for toroidal
and projective-planar graphs containing no K33-subdivisions, we apply these
techniques to obtain their unlabelled enumeration.Comment: 25 pages (some corrections), 4 figures (one figure added), 3 table
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
We describe the structure of 2-connected non-planar toroidal graphs with no
K_{3,3}-subdivisions, using an appropriate substitution of planar networks into
the edges of certain graphs called toroidal cores. The structural result is
based on a refinement of the algorithmic results for graphs containing a fixed
K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing
K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these
graphs in linear-time and makes possible to enumerate labelled 2-connected
toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex
degree two or three by using an approach similar to [A. Gagarin, G. Labelle,
and P. Leroux, "Counting labelled projective-planar graphs without a
K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table
The obstructions for toroidal graphs with no 's
Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no -subdivisions that coincide with
the toroidal graphs with no -minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no 's and prove that the
lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail
Multiple domination models for placement of electric vehicle charging stations in road networks
Electric and hybrid vehicles play an increasing role in the road transport
networks. Despite their advantages, they have a relatively limited cruising
range in comparison to traditional diesel/petrol vehicles, and require
significant battery charging time. We propose to model the facility location
problem of the placement of charging stations in road networks as a multiple
domination problem on reachability graphs. This model takes into consideration
natural assumptions such as a threshold for remaining battery load, and
provides some minimal choice for a travel direction to recharge the battery.
Experimental evaluation and simulations for the proposed facility location
model are presented in the case of real road networks corresponding to the
cities of Boston and Dublin.Comment: 20 pages, 5 figures; Original version from March-April 201
Classification of finite groups with toroidal or projective-planar permutability graphs
Let be a group. The permutability graph of subgroups of , denoted by
, is a graph having all the proper subgroups of as its vertices,
and two subgroups are adjacent in if and only if they permute. In
this paper, we classify the finite groups whose permutability graphs are
toroidal or projective-planar. In addition, we classify the finite groups whose
permutability graph does not contain one of , , , ,
or as a subgraph.Comment: 30 pages, 8 figure
The complement of proper power graphs of finite groups
For a finite group , the proper power graph of is
the graph whose vertices are non-trivial elements of and two vertices
and are adjacent if and only if and or for some
positive integer . In this paper, we consider the complement of
, denoted by . We classify all
finite groups whose complement of proper power graphs is complete, bipartite, a
path, a cycle, a star, claw-free, triangle-free, disconnected, planar,
outer-planar, toroidal, or projective. Among the other results, we also
determine the diameter and girth of the complement of proper power graphs of
finite groups.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes
have been mad
- …