18 research outputs found
Circumference and Pathwidth of Highly Connected Graphs
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t
has treewidth at most t-1. Under the additional assumption of 2-connectivity,
such graphs have bounded pathwidth, which is a qualitatively stronger result.
Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007]
who showed that every graph without k disjoint cycles of length at least t has
bounded treewidth (as a function of k and t). Our main result states that,
under the additional assumption of (k + 1)- connectivity, such graphs have
bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover,
examples show that (k + 1)-connectivity is required for bounded pathwidth to
hold. These results suggest the following general question: for which values of
k and graphs H does every k-connected H-minor-free graph have bounded
pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure
The L(2,1)-labeling of unigraphs
The L(2, 1)-labeling problem consists of assigning colors from the integer set 0 ...., lambda to the nodes of a graph G in such a way that nodes at a distance of at most two get different colors, while adjacent nodes get colors which are at least two apart. The aim of this problem is to minimize lambda and it is in general NP-complete. In this paper the problem of L(2, 1)-labeling unigraphs, i.e. graphs uniquely determined by their own degree sequence up to isomorphism, is addressed and a 3/2-approximate algorithm for L(2, 1)-labeling unigraphs is designed. This algorithm runs in 0(n) time, improving the time of the algorithm based on the greedy technique, requiring 0(m) time, that may be near to Theta (n(2)) for unigraphs. (C) 2011 Elsevier B.V. All rights reserved
Decomposition, approximation, and coloring of odd-minor-free graphs
We prove two structural decomposition theorems about graphs excluding
a fixed odd minor H, and show how these theorems can
be used to obtain approximation algorithms for several algorithmic
problems in such graphs. Our decomposition results provide new
structural insights into odd-H-minor-free graphs, on the one hand
generalizing the central structural result from Graph Minor Theory,
and on the other hand providing an algorithmic decomposition
into two bounded-treewidth graphs, generalizing a similar result for
minors. As one example of how these structural results conquer difficult
problems, we obtain a polynomial-time 2-approximation for
vertex coloring in odd-H-minor-free graphs, improving on the previous
O(jV (H)j)-approximation for such graphs and generalizing
the previous 2-approximation for H-minor-free graphs. The class
of odd-H-minor-free graphs is a vast generalization of the well-studied
H-minor-free graph families and includes, for example, all
bipartite graphs plus a bounded number of apices. Odd-H-minor-free
graphs are particularly interesting from a structural graph theory
perspective because they break away from the sparsity of H-
minor-free graphs, permitting a quadratic number of edges
On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs
Numerous problems consisting in identifying vertices in graphs using
distances are useful in domains such as network verification and graph
isomorphism. Unifying them into a meta-problem may be of main interest. We
introduce here a promising solution named Distance Identifying Set. The model
contains Identifying Code (IC), Locating Dominating Set (LD) and their
generalizations -IC and -LD where the closed neighborhood is considered
up to distance . It also contains Metric Dimension (MD) and its refinement
-MD in which the distance between two vertices is considered as infinite if
the real distance exceeds . Note that while IC = 1-IC and LD = 1-LD, we have
MD = -MD; we say that MD is not local
In this article, we prove computational lower bounds for several problems
included in Distance Identifying Set by providing generic reductions from
(Planar) Hitting Set to the meta-problem. We mainly focus on two families of
problem from the meta-problem: the first one, called bipartite gifted local,
contains -IC, -LD and -MD for each positive integer while the
second one, called 1-layered, contains LD, MD and -MD for each positive
integer . We have:
- the 1-layered problems are NP-hard even in bipartite apex graphs,
- the bipartite gifted local problems are NP-hard even in bipartite planar
graphs,
- assuming ETH, all these problems cannot be solved in when
restricted to bipartite planar or apex graph, respectively, and they cannot be
solved in on bipartite graphs,
- even restricted to bipartite graphs, they do not admit parameterized
algorithms in except if W[0] = W[2]. Here is the
solution size of a relevant identifying set.
In particular, Metric Dimension cannot be solved in under ETH,
answering a question of Hartung in 2013