An unsatisfiable set is a set of formulas whose conjunction is unsatisfiable. Every unsatisfiable set can be corrected, i.e., made satisfiable, by removing a subset of its members. The subset whose removal yields satisfiability is called a correction subset. Given an unsatisfiable set F there is a well known hitting set duality between the unsatisfiable subsets of F and the correction subsets o...

Let G be a simple graph of order n. The path cover number μ(G) is defined to be the minimum number of vertex disjoint paths required to cover the vertices of G. Ore proved that in general μ(G) ≤ max{1, n − σ2(G)}. We conjecture that if G is k-regular, then μ(G) ≤ n k+1 and we prove this for k ≤ 5.

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerpla...

Glacial forelands are extremely sensitive to temperature changes and are therefore appropriate places to explore the development of microbial communities in response to climate-driven deglaciation. In this study, we investigated the bacterial communities that developed at the initial stage of deglaciation using space-for-time substitution in the foreland of an ice sheet in Larsemann Hills. A se...

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a c...

We consider a planning problem that generalizes Alcuin’s river crossing problem to scenarios with arbitrary conflict graphs. This generalization leads to the so-called Alcuin number of the underlying conflict graph. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around the Alcuin number. Our technical main result is an NP-certificate for the ...

Governments around the world are increasingly devolving authority for forest management to the local level in an attempt to strengthen the management of national forests. Community forestry programs are recognized as providing a range of economic and social benefits and having a positive impact on increasing forest cover. However, concerns have been raised about the capability of user groups to...

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected n-vertex graph has a path cover with at most dn/2e paths. We prove Gallai’s conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most b5n/8c paths, which is an improvement over the best previously known b...