Fault-free Hamiltonian cycle including given edges in folded hypercubes with faulty edges

The folded hypercube is an important interconnection network for multiprocessor systems. Let [Formula: see text] with [Formula: see text] denote an [Formula: see text]-dimensional folded hypercube. For a given fault-free edge set [Formula: see text] with [Formula: see text] and a faulty edge set [Formula: see text] with [Formula: see text], in this paper we prove that [Formula: see text] contains a fault-free Hamiltonian cycle including each edge of [Formula: see text] if and only if the subgraph induced by [Formula: see text] is linear forest. Furthermore, we give the definitions of the distance among three vertex-disjoint edges and the distance between a vertex and a vertex set. For three vertex-disjoint edges [Formula: see text], the distance among them is denoted by [Formula: see text]. For a vertex [Formula: see text] and a vertex set [Formula: see text], the distance between [Formula: see text] and [Formula: see text] is denoted by [Formula: see text].

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on $$(sP_1+ P_3)$$ ( s P 1 + P 3 ) -free graphs for every integer $$s\ge 1$$ s ≥ 1 . We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are -complete on $$(P_2+ P_5,P_6)$$ ( P 2 + P 5 , P 6 ) -free graphs.

Prescribed Hamiltonian Connectedness of 2D Torus

Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

High Precision Altimeter Demonstrates Simplification and Depression of Microtopography on Seismic Lines in Treed Peatlands

Seismic lines are linear forest clearings used for oil and gas exploration. The mechanical opening of forests for these narrow (3–10 meter) lines is believed to simplify microtopographic complexity and depress local topographic elevation. In treed peatlands, simplified microtopography limits tree regeneration by removing favourable microsites (hummocks) for tree recruitment and increasing the occurrence of flooding that reduces survival of tree seedlings. Little, however, has been done to quantify the microtopography of seismic lines and specifically the degree of alteration. Here, we measured microtopography at 102 treed peatland sites in northeast Alberta, Canada using a high precision hydrostatic altimeter (ZIPLEVEL PRO-2000) that measured variation in local topography of seismic lines and adjacent paired undisturbed forests. Sites were separated into four peatland ecosite types and the presence or absence of recent (<22 years) wildfires. Paired t-tests were used to compare microtopographic complexity and depression depth of seismic lines compared with adjacent forests. Microtopographic complexity on seismic lines was simplified by 20% compared to adjacent stands with no significant change between recently burned and unburned sites, nor between ecosites. Not only were seismic lines simplified, but they were also depressed in elevation by an average of 8 cm compared to adjacent forests with some minor variation between ecosites observed, but again not with recent wildfires. Thus, simplification of microtopographic complexity and the creation of depressions can persist decades after initial disturbance with some differences between peatland ecosites, implying the need for ecosite-specific restoration of topographic complexity. The importance of microtopography for tree regeneration on seismic lines remains an important question for reforestation of these disturbances and thus long-term recovery of habitat for species dependent on undisturbed peatlands, including woodland caribou.

Aboveground biomass estimation in linear forest objects: 2D- vs. 3D-data

Wood-chips of linear forest objects (hedge banks and roadside plantings) are used as sustainable energy supply in wood-chip heating systems. However, wood yield of linear forest objects is very heterogeneous and hard to estimate in advance. The aim of the present study was to compare the dry mass estimation potentials of two different non-destructive data: (i) Canopy area (derived from aerial images) and mean age at stump level (2D), (ii) volume of vegetation cover based on structure from motion (SfM) via unmanned aerial vehicle (3D). These two types of data were separately used to predict reference dry mass (ground truth) in eleven objects (5 hedge banks and 6 roadside plantings) in Schleswig-Holstein, Germany. The predicting potentials were compared afterwards. The reference dry mass was ascertained by weighing after harvesting and drying samples to constant weight. The model predicting reference dry mass using canopy area and mean age at stump level achieved a relative root mean square error (RMSE) of 52% (42% at larger combined plot sizes). The model predicting reference dry mass using SfM volume achieved a relative RMSE of 30% (16% at larger combined plot sizes). This result indicates that biomass is better described by volume of vegetation cover than by canopy area and age.