21 research outputs found
Chinese Postman Problem on edge-colored multigraphs
It is well-known that the Chinese postman problem on undirected and directed
graphs is polynomial-time solvable. We extend this result to edge-colored
multigraphs. Our result is in sharp contrast to the Chinese postman problem on
mixed graphs, i.e., graphs with directed and undirected edges, for which the
problem is NP-hard
Self-Assembly of DNA Graphs and Postman Tours
DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and the Windy Postman Problem
An updated annotated bibliography on arc routing problems
The number of arc routing publications has increased significantly in the last decade. Such an increase justifies a second annotated bibliography, a sequel to Corberán and Prins (Networks 56 (2010), 50–69), discussing arc routing studies from 2010 onwards. These studies are grouped into three main sections: single vehicle problems, multiple vehicle problems and applications. Each main section catalogs problems according to their specifics. Section 2 is therefore composed of four subsections, namely: the Chinese Postman Problem, the Rural Postman Problem, the General Routing Problem (GRP) and Arc Routing Problems (ARPs) with profits. Section 3, devoted to the multiple vehicle case, begins with three subsections on the Capacitated Arc Routing Problem (CARP) and then delves into several variants of multiple ARPs, ending with GRPs and problems with profits. Section 4 is devoted to applications, including distribution and collection routes, outdoor activities, post-disaster operations, road cleaning and marking. As new applications emerge and existing applications continue to be used and adapted, the future of arc routing research looks promising.info:eu-repo/semantics/publishedVersio
Decomposition of a graph into two disjoint odd subgraphs
An odd (resp. even) subgraph in a multigraph is its subgraph in which every
vertex has odd (resp. even) degree. We say that a multigraph can be decomposed
into two odd subgraphs if its edge set can be partitioned into two sets so that
both form odd subgraphs. In this paper we give a necessary and sufficient
condition for the decomposability of a multigraph into two odd subgraphs. We
also present a polynomial time algorithm for finding such a decomposition or
showing its non-existence. We also deal with the case of the decomposability
into an even subgraph and an odd subgraph
Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours
We present a new approach for gluing tours over certain tight, 3-edge cuts.
Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles
in special graph classes and in proving bounds for 2-edge-connected subgraph
problem, but not much was known in this direction for gluing connected
multigraphs. We apply this approach to the traveling salesman problem (TSP) in
the case when the objective function of the subtour elimination relaxation is
minimized by a -cyclic point: ,
where the support graph is subcubic and each vertex is incident to at least one
edge with -value 1. Such points are sufficient to resolve TSP in general.
For these points, we construct a convex combination of tours in which we can
reduce the usage of edges with -value 1 from the of
Christofides algorithm to while keeping the
usage of edges with fractional -value the same as Christofides algorithm. A
direct consequence of this result is for the Uniform Cover Problem for TSP: In
the case when the objective function of the subtour elimination relaxation is
minimized by a -uniform point: , we
give a -approximation algorithm for TSP. For such points, this
lands us halfway between the approximation ratios of of
Christofides algorithm and implied by the famous "four-thirds
conjecture"
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio