An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti

Abstract

Let G = (V, E) be a nonempty graph and ξ : E → N be a function. In the paper we study the computational complexity of the problem of finding vertex colorings c of G such that:We show that the problem is NP-hard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for cycles and bipartite graphs. Next, we use the last two results to construct an algorithm that solves the problem for a given cactus G in O(n log n) time, where n is the number of vertices of G.

Microbial mats and their palaeoenvironmental analysis in offshore – shelf facies of the Los Molles Formation (Toarcian – Lower Callovian) in the Chacay Melehue area, Neuquén Basin, Argentina

“…As previously mentioned, the Los Molles Formation has been interpreted as deposited in an offshore to shelf environment affected by storms and density currents (Campetella et al, 2020;Ponce et al, 2015), and offshore to prodelta (Zavala et al, 2014(Zavala et al, , 2020, with calm waters and low energy, with depths ranging between 200 and 400 m (Arregui et al, 2011;Gomez Omil et al, 2002). All the levels with microbial mats, including those with Kinneyia structures, were found in deposits formed in an offshore to shelf environment (Campetella et al, 2020;Ponce et al, 2015;Rodriguez et al, 2018), which would have been ideal for the development and establishment of the bacterial communities (Noffke, 2009;Wacey, 2009).…”

“…These ichnogenera occur in clusters and in restricted intervals, representing an ichnoassemblage with low-to-moderate abundance and low diversity of trace fossils (Campetella et al, 2020). Subordinately, tt the base and the middle part of the section, turbiditic deposits integrated by large-scale slumps, thick debris-flow deposits, and channel and levee geometries with abundant phytodetrital material were recognized (Campetella et al, 2020;Ponce et al, 2015), and interpreted as intrabasinal and extrabasinal turbidites (sensu Zavala & Arcuri, 2016). In previous studies, these deposits were interpreted as density-currents deposits, large-scale slumps, and subaquatic lahars (Gulisano & Gutierrez Pleimling, 1994;.…”

“…Argentina has six oil basins that show different levels of hydrocarbon production (Uliana et al, 2014). The Neuquén Basin stands out of these basins, as it contributes 42% of oil and 55% of gas production in the country, consolidating itself as the most important basin in Argentina (Ponce et al, 2015). The Los Molles, Vaca Muerta and Agrio formations were deposited in marine environments and act as hydrocarbon-generating rocks in the Neuquén Basin (Legarreta & Villar, 2012).…”

“…Although there are mentions of fossil microbial mats in some formations in the Neuquén Basin (Comerio et al, 2014;Kietzmann et al, 2011;Paz et al, 2021;Tunik et al, 2009), those studies are mostly focused on a regional scale. For this reason, the importance of this contribution is that it represents the first study focused particularly on the description and characterization of the fossil microbial mats in the Los Molles Formation, in offshore to shelf environments, affected by storms, and density currents (Campetella et al, 2020;Ponce et al, 2015).…”

Edge coloring of graphs of signed class 1 and 2

“…Clearly every tree is a cactus as it doesn't have cycles at all. It is a well known fact [6] that graph G is a cactus if and only if there exists a sequence of graphs G 1 , . .…”