11,344 research outputs found
On spanning tree congestion
We prove that every connected graph G of order n has a spanning tree T such that for every edge e of T the edge-cut defined in G by the vertex sets of the two components of T - e contains at most n^{\frac{3}{2}} many edges which solves a problem posed by Ostrovskii (Minimal congestion trees, Discrete Math. 285 (2004), 219-226.
Treewidth and related graph parameters
For modeling some practical problems, graphs play very important roles.
Since many modeled problems can be NP-hard in general, some restrictions
for inputs are required. Bounding a graph parameter of the inputs is one of
the successful approaches. We study this approach in this thesis. More precisely,
we study two graph parameters, spanning tree congestion and security
number, that are related to treewidth.
Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T),
the congestion of e is the number of edges in G connecting two components
of T − e. The edge congestion of G in T is the maximum congestion over all
edges in T. The spanning tree congestion of G is the minimum congestion
of G in its spanning trees. In this thesis, we show the spanning tree congestion
for the complete k-partite graphs, the two-dimensional tori, and the twodimensional
Hamming graphs. We also address lower bounds of spanning
tree congestion for the multi-dimensional hypercubes, the multi-dimensional
grids, and the multi-dimensional Hamming graphs.
The security number of a graph is the cardinality of a smallest vertex subset
of the graph such that any “attack” on the subset is “defendable.” In this thesis,
we determine the security number of two-dimensional cylinders and tori.
This result settles a conjecture of Brigham, Dutton and Hedetniemi [Discrete
Appl. Math. 155 (2007) 1708–1714]. We also show that every outerplanar
graph has security number at most three. Additionally, we present lower and
upper bounds for some classes of graphs.学位記番号:工博甲39
Better Hardness Results for the Minimum Spanning Tree Congestion Problem
In the spanning tree congestion problem, given a connected graph , the
objective is to compute a spanning tree in for which the maximum edge
congestion is minimized, where the congestion of an edge of is the
number of vertex pairs adjacent in for which the path connecting them in
traverses . The problem is known to be NP-hard, but its approximability
is still poorly understood, and it is not even known whether the optimum can be
efficiently approximated with ratio . In the decision version of this
problem, denoted STC-, we need to determine if has a spanning tree with
congestion at most . It is known that STC- is NP-complete for ,
and this implies a lower bound of on the approximation ratio of
minimizing congestion. On the other hand, -STC can be solved in polynomial
time, with the complexity status of this problem for
remaining an open problem. We substantially improve the earlier hardness result
by proving that STC- is NP-complete for . This leaves only the case
open, and improves the lower bound on the approximation ratio to
Minimum congestion spanning trees in planar graphs
The main purpose of the paper is to develop an approach to evaluation or
estimation of the spanning tree congestion of planar graphs. This approach is
used to evaluate the spanning tree congestion of triangular grids
Parameterized complexity of the spanning tree congestion problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.publishedVersio
Complexity Results for the Spanning Tree Congestion Problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k
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