Given an undirected graph G and an integer k ≥ 0, the NPhard 2-Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum feedback edge set is a natural parameter with f ≤ k. We improve on previous fixed-parameter tract...

We consider the problems of finding a caterpillar tree in a graph. We first prove that, unless P=NP, there is no approximation algorithms for finding a minimum spanning caterpillar in a graph within a factor of f(n); where f(n) is any polynomial time computable function of n, the order of the graph. Then we present a quadratic integer programming formulation for the problem that can be a base f...

We characterize two types of bar-visibility graphs on the Mobius band (abbreviated "BVGMs"), in which vertices correspond to intervals that are parallel or orthogonal to the axis of the band, depen­ ding on type, and in which adjacency corresponds to orthogonal visibility of intervals. BVGMs with intervals orthogonal to the axis are shown to be equivalent to the "polar visibility graphs" studie...

The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4], [5]) states that for any positive integer n (except numbers from a given 49 element set), one can find a tree with Wiener index n. In this paper, we prove that every i...

We time-studied a Caterpillar 950F front-end loader and a Caterpillar 528 grapple skidder used to extract bunched whole trees to a landing in a short rotation Eucalyptus plantation. The loader was 40-60% more productive than the grapple skidder, depending on extraction distance. Alternatively, the single loader could both extract trees and handle the landing duties, such as moving residues from...

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y a convex graph, if there an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ consecutive with respect to the X, G said have convexity X. k-star caterpillar tree collection stars each star having k vertices degree one roots are joined by path. For partitions Y, we associate on in its neighborhood...

OBJECTIVE To describe previously unreported oropharyngeal manifestations and management of caterpillar ingestion. DESIGN Retrospective case series. SETTING Tertiary children's hospital. PATIENTS A total of 733 cases of exposure to caterpillars from January 1, 1994, to November 1, 1997, were reviewed. Twenty-six patients had oropharyngeal exposure with 8 patients ingesting the caterpillar....

Let $$\Omega _n$$ be the family of binary trees on n vertices obtained by identifying root an rgood tree with a vertex maximum eccentricity caterpillar. In paper titled “On different middle parts tree" (The Electronic Journal Combinatorics, 25 (2018), no. 3, 3.17, 32 pp), Smith et al. conjectured that among all pairwise distance between any two center, centroid and subtree core is maximized som...

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number γ s (G) and the signed total 2-domination number γ st(G) of a graph G are variants of the signed domination number γs(G) and the signed total domination number γst(G). Their values for caterpillars are studied.

By creating or modifying habitats used by other organisms, physical ecosystem engineers can influence local patterns of biological diversity. However, there have been very few empirical studies quantifying engineering effects in different biological systems. In this study, we examined the effect of shelter-building caterpillars on the species richness and guild structure of leaf-chewing herbivo...