48,781 research outputs found
The Minimum Shared Edges Problem on Grid-like Graphs
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide
whether it is possible to route paths from a start vertex to a target
vertex in a given graph while using at most edges more than once. We show
that MSE can be decided on bounded (i.e. finite) grids in linear time when both
dimensions are either small or large compared to the number of paths. On
the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known
that MSE is fixed-parameter tractable with respect to the number of paths.
We show that, under standard complexity-theoretical assumptions, the problem
parametrised by the combined parameter , , maximum degree, diameter, and
treewidth does not admit a polynomial-size problem kernel, even when restricted
to planar graphs
On the diameter of random planar graphs
We show that the diameter D(G_n) of a random labelled connected planar graph
with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there
exists a constant c>0 such that the probability that D(G_n) lies in the
interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than
1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We
prove similar statements for 2-connected and 3-connected planar graphs and
maps.Comment: 24 pages, 7 figure
Diameter and Treewidth in Minor-Closed Graph Families
It is known that any planar graph with diameter D has treewidth O(D), and
this fact has been used as the basis for several planar graph algorithms. We
investigate the extent to which similar relations hold in other graph families.
We show that treewidth is bounded by a function of the diameter in a
minor-closed family, if and only if some apex graph does not belong to the
family. In particular, the O(D) bound above can be extended to bounded-genus
graphs. As a consequence, we extend several approximation algorithms and exact
subgraph isomorphism algorithms from planar graphs to other graph families.Comment: 15 pages, 12 figure
The degree/diameter problem in maximal planar bipartite graphs
The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D)
and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
Diameter Bounds for Planar Graphs
The inverse degree of a graph is the sum of the reciprocals of the degrees of
its vertices. We prove that in any connected planar graph, the diameter is at
most 5/2 times the inverse degree, and that this ratio is tight. To develop a
crucial surgery method, we begin by proving the simpler related upper bounds
(4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which
are also tight
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
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