79 research outputs found
Hamiltonicity below Dirac's condition
Dirac's theorem (1952) is a classical result of graph theory, stating that an
-vertex graph () is Hamiltonian if every vertex has degree at
least . Both the value and the requirement for every vertex to have
high degree are necessary for the theorem to hold.
In this work we give efficient algorithms for determining Hamiltonicity when
either of the two conditions are relaxed. More precisely, we show that the
Hamiltonian cycle problem can be solved in time , for some
fixed constant , if at least vertices have degree at least , or
if all vertices have degree at least . The running time is, in both
cases, asymptotically optimal, under the exponential-time hypothesis (ETH).
The results extend the range of tractability of the Hamiltonian cycle
problem, showing that it is fixed-parameter tractable when parameterized below
a natural bound. In addition, for the first parameterization we show that a
kernel with vertices can be found in polynomial time
Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories
We survey results on Hedetniemi’s conjecture which are connected to adjoint functors in the “thin” category of graphs, and expose the obstacles to extending these results
Towards integrated care in breastfeeding support: a cross-sectional survey of practitioners’ perspectives
Second-order analysis of space-time disease clustering
We consider the problem of detecting and describing space-time interaction in point process data. We extend existing second-order methods for purely spatial point process data to the spatial-temporal setting. This extension allows us to estimate space-time interaction as a function of spatial and temporal separation, and provides a useful reinterpretation of a popular test, due to Knox, for space-time interaction. Applications to simulated and real data indicate the method's potential
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