64 research outputs found
Unexpected behaviour of crossing sequences
The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number
of crossings in a drawing of G on an orientable surface of genus n. We prove
that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) =
b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al.
and resolves a problem of Salazar.Comment: 21 page
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
Tree-based decompositions of graphs on surfaces and applications to the traveling salesman problem
The tree-width and branch-width of a graph are two well-studied examples of parameters that measure how well a given graph can be decomposed into a tree structure. In this thesis we give several results and applications concerning these concepts, in particular if the graph is embedded on a surface.
In the first part of this thesis we develop a geometric description of tangles in graphs embedded on a fixed surface (tangles are the obstructions for low branch-width), generalizing a result of Robertson and Seymour. We use this result to establish a relationship between the branch-width of an embedded graph and the carving-width of an associated graph, generalizing a result for the plane of Seymour and Thomas. We also discuss how these results relate to the polynomial-time algorithm to determine the branch-width of planar graphs of Seymour and Thomas, and explain why their method does not generalize to surfaces other than the sphere.
We also prove a result concerning the class C_2k of minor-minimal graphs of branch-width 2k in the plane, for an integer k at least 2.
We show that applying a certain construction to a class of graphs in the projective plane yields a subclass of C_2k, but also show that not all members of C_2k arise in this way if k is at least 3.
The last part of the thesis is concerned with applications of graphs of bounded tree-width to the Traveling Salesman Problem (TSP). We first show how one can solve the separation problem for comb inequalities (with an arbitrary number of teeth) in linear time if the tree-width is bounded. In the second part, we modify an algorithm of Letchford et al. using tree-decompositions to obtain a practical method for separating a different class of TSP inequalities, called simple DP constraints, and study their effectiveness for solving TSP instances.Ph.D.Committee Chair: Thomas, Robin; Committee Co-Chair: Cook, William J.; Committee Member: Dvorak, Zdenek; Committee Member: Parker, Robert G.; Committee Member: Yu, Xingxin
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
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