14 research outputs found
How tough is toughness?
The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors
Some local--global phenomena in locally finite graphs
In this paper we present some results for a connected infinite graph with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of . (For a vertex
of a graph the ball of radius centered at is the subgraph of
induced by the set of vertices whose distance from does not
exceed ). In particular, we prove that if every ball of radius 2 in is
2-connected and satisfies the condition for
each path in , where and are non-adjacent vertices, then
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in satisfies Ore's
condition (1960) then all balls of any radius in are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
A look at cycles containing specified elements of a graph
AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration
Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number
A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle,
respectively) that traverses all of its vertices. The problems of deciding
their existence in an input graph are well-known to be NP-complete, in fact,
they belong to the first problems shown to be computationally hard when the
theory of NP-completeness was being developed. A lot of research has been
devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems
for special graph classes, yet only a handful of positive results are known.
The complexities of both of these problems have been open even for -free
graphs, i.e., graphs of independence number at most . We answer this
question in the general setting of graphs of bounded independence number.
We also consider a newly introduced problem called
\emph{Hamiltonian--Linkage} which is related to the notions of a path
cover and of a linkage in a graph. This problem asks if given pairs of
vertices in an input graph can be connected by disjoint paths that altogether
traverse all vertices of the graph. For , Hamiltonian-1-Linkage asks
for existence of a Hamiltonian path connecting a given pair of vertices. Our
main result reads that for every pair of integers and , the
Hamiltonian--Linkage problem is polynomial time solvable for graphs of
independence number not exceeding . We further complement this general
polynomial time algorithm by a structural description of obstacles to
Hamiltonicity in graphs of independence number at most for small values of
Twin-constrained Hamiltonian paths on threshold graphs: an approach to the minimum score separation problem
The Minimum Score Separation Problem (MSSP) is a combinatorial problem that has been introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting-stock problem. During the process of producing boxes, áat papers are prepared for folding by being scored with knives. The problem is to determine if and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. While it was originally suggested to analyse the MSSP as a specific variant of a Generalized Travelling Salesman Problem, the thesis introduces the concept of twin-constrained Hamiltonian cycles and models the MSSP as the problem of finding a twin-constrained Hamiltonian path on a threshold graph (threshold graphs are a specific type of interval graphs).
For a given undirected graph G(N,E) with an even node set N and edge set E, and a bijective function b on N that assigns to every node i in N a "twin node" b(i)6=i, we define a new graph G'(N,E') by adding the edges {i,b(i)} to E. The graph G is said to have a twin-constrained Hamiltonian path with respect to b if there exists a Hamiltonian path on G' in which every node has its twin node as its predecessor (or successor).
We start with presenting some general Öndings for the construction of matchings, alternating paths, Hamiltonian paths and alternating cycles on threshold graphs. On this basis it is possible to develop criteria that allow for the construction of twin-constrained Hamiltonian paths on threshold graphs and lead to a heuristic that can quickly solve a large percentage of instances of the MSSP. The insights gained in this way can be generalized and lead to an (exact) polynomial time algorithm for the MSSP. Computational experiments for both the heuristic and the polynomial-time algorithm demonstrate the efficiency of our approach to the MSSP. Finally, possible extensions of the approach are presented
Chvátal-Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs
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